Upload
cher-azer
View
27
Download
1
Embed Size (px)
Citation preview
@ò î ö b — y ⁄ a @t ì z j Û a ë @k í Š ‡ n Ü Û @ï i ‹ È Û a @‡ è È ¾ a@
�������� ����
��������� ����� �� ���� ������ ������
� � ���� �� �� � � � �� � �� � �� � ��� � � ��� � � � � �� �� � � ��
1
��������������������
������ � �� ���� ��������� ��� ���� �� ������ ����� � � � � � � � �� ��� � �� � ��� � � �! ��� " ����# $ � �� � �% � & ���
��� �! ��� � � �� �� � ' � � �� ������ � � �( ��� ' �) � ' �( ���. � " �+ � , � - . � , � �/ �� � 0 �. � �� �1 � � 2 �� 3 � � �+ � ��(
4 ��� , � / 5 � � �� ���� ���� � ' 6 �7 �� " 1 �� � 8 �9 �� �1 � : �� 2 � �� �� ��� 5 , �) , � 6 ��� �� � � 7 ' , ���� ( � ; ��' 0 �. � �� � ' � + � ' � ��� �� < ( = � �� , � � ��� > 1 � � = �( �� , � � �� 7 �� � ( ? � -
�� � > � @ � � �� �� � 6 �� � �� & �� �� . ( 8 �9 �� �1 � � ��# � �1 � �� 8 � ���� < ���� �� ���� ��������� % ��5 ��� ��� A � ��� � �� � � ��# � ��� �����
�������� > 1 � 8 ���) � �A ���� , ������ - # ���� .( + �� 7 �� ��������� , + 6 + � ! < �D ' �. D � � ��� �� - ����� < ( �) � ��� - ( �� ���
�� ��( �� �� � �� , � � ���. �� ����� �� ��� � �� � �.
� � � �
.����� �� ���� �
2
� � � � � � � � � � � � �
3
� � � � � � � � � � � � � � � � � � � �� � �
PROBABILITIES
� ��������� ��� ������ �� � � � ��� ������� ��� �� �� " � � � � � �
� � � � � � � � � � � �(E v e n t )� � � � . " � � � � �! �� ���� " ��# ! $ ���� �% ��� & ' �����% �� (�) !
(* ) � ���% '�! ���� �� ���� + ����� ��� �� ������� �� ��� �� �, �� & % �% � " �- �� & ' $ ����� . ��/ �� 0 1 % � $ ���2 � ���� �% � $ ����� $ ����� . ��/ ��
+ ����� " ��� �� ��' 3 � & ��� $ ���� . 4�� ������ �� ���% �� % � 5 �, ! $ - �"& ���� " �. � 4�# �� �� �� � ��� $ ��� � ��)52 (� �. $ - 4/1
�� $ ��� � �� � ��� $ - � �% ��� �� �, 9 �% % � �. $ ����� & ' : % �% �� �; ' & ' & ���� $ - � < # � �= �% �� > - ��% �% % ? ' " �� � � 4�# ��4/1 " ��
���� � �@ 5 �, � 4����"& ������ " �. $ ��� ��� & '13 . �. �������' �, 9 52/13 = 4/1.
2 � 0 ���� �% ��� �2 �� B �% �� �, �� (�� �� �����. � 2 � > ������ �� , 9 ��1 �� 5 # � & ' 5 �, � $ ���� ��/ �� " % �! 3 � 3 ������ " ��� �� �� �= - & ' �2 �% � " % �! % �� � & �� 4�# �� ��� ������� 4���� & ' $ �# ��� $ C �% �� ��' D , . " % �! � � ��% ' & ' $ = �E ����� & ' �' ����
�� � & ' $ = �E � ����� 0 �� < �9 ��� ��� �������� �2 - � � " �C ��� ���E �� �� 4�! �� & ' ��= ' 4��� < # � �. ���� $ �# � � + ��# � < �9 ��? # '
��! �� ��) �� " �C ���(PASCAL ) " ��' � (F E R M AT ) & ��% ��
4
(BE R N O U L L I ) $ �� " ��� �� & ' + ��� $ ��% " % �! �% . � �. �� ������� �# � & '.
� ���� � ���: � �� (��� �% �� # ������� �# � " �/ % � 0 �� F � �� �! ��% & !
�� ��� �, . & ' � E ��� & ��� ������ F ��: 1� � � � � � � � � � � � � � � � � � � � � � � � � � � :
�! / �* % � (* ) � % �� �. � & �! �� $ � � + � ; � ���% �% % � F �' � $ ���� : �@ � �� ��/ & . � $ % ! ���� H I ��% ��1 �� 2 �� 3 �� 4 �� 5
��6 J � J ' �- ��/ � ������ �% % � F �1 % � 1 �� 3 �� 5 �K� $ KK� ��� . < �KK�� �KK. � �� & KK� $ KK# �� �; KK' �, KK! . �� � � � �
(E x p e r i m e n t ) < �� �% ����. � ��� �. J ' �- ��/ � !� " � � � (E v e n t ) < �K�� ��/ �� $ % ! ���� " ����� > � $ ��� �� � � � � � � �
� � � � � � .(S a m p l e S p a c e ) / �* � & % ��� L �1 �� �� ) ! � �� $ ��� ��! - 3 ���� �� .
2� � � � � � � � # � � � � �(P o s s i b l e C a s e s ) + � M $ �% �/ � �� �! � & ��� $ 1 # �E ��� H I ��% �� �� " ����� & .
$ % �� $ � �� �� ��= ��� �% ��! � $ # �� $ ��- & � % � (* ) �' ��� �% ��! � % �. � & � % �� � $ ���!1 �� 2 �� 3 �� 4 �� 5
��6 $ % ! ���� " ����� � �� ���' 2 � $ # ���� $ ��- & � $ ��� & ' 6 % �� �. � & � $ ��� & '.
3� � � � � � � � � # � � � � �(F a v o r a b l e C a s e s )
5
�. J , �� 3 ���� ���� < �9 J N � & ��� " ����� �� H I ��% �� & . & ' J ' �- < # � ��= ��� �. 3 ���� ��! �, ; ' � �% ����. � > C ��
$ ��� & . 3 ���� �, . ���� & ��� " ����� �; ' % �� �. � & � < # � ��= ���1 �� 3 �� 5 � " ����� < ��� $ ) * ) �� " ����� D , .
$ ������. 4� � $ " � � � � � � # � � � � �(E q u a l l y L i k e l y C a s e s )
& ' $ �% � �� � ��� � �� �� $ ��% = � $ % �� " �! � � �% � ��! �, 9% �C � � ���� �� ��� 0 1 % ���� $ ' �) ! �� ��% � �! �% ���� 0 ! & ' �. �
�! � ��! J � $ # ) ���� " ��� ��! � " �! �� D , . �; ' (� ���# E ��4���� & ' 4= % �� 0 1 % ��% �.
5� � � % � � � � � � � � � � � �(M u t u a l l y E x c l u s i v e E v e n t s ) �) ���� �� ���A � B ��� ���) � � ������ �, 9 ��' �% �� ���% � .
! � � % �� � & � % � (* ) �' " - � & ' �� � < # � ��= ��� � ���.
6� � $ � � & � � � � � � � � �(I n d e p e n d e n t E v e n t s ) �) ���� ���A �� B � � �� ��. � �9 B �- � ��! �, 9 �# ���� �) ��
& ' ) N � : ��- � E O � B �- �. � ��� $ # �� $ ��- & � % � (* ) �'�@ � $ �% � ) ? �� � $ % �) �� $ ��� $ �% �; ' ������� ���< �.
7� � $ � � ' � � � � � � � �(E x h a u s t i v e E v e n t s ) 3 ����� < ���A � B � C ... ��! �, 9 �� $ � � & ' $ # ��2 3 ���
$ � ��� + � 9 % � �. � �9 3 � � �� ��.
6
��! �, 9 �� : ���� $ ' ��� $ ��� �� �� 4��� ��E � % � (* ) �'� �� : % @ $ # ��2 3 ��� " ����� D , . ���� �E � � �� �% E � 1 #
" �1 = �� D , . �� � ��� $ 1 = : � ��! �� . < # � ��= ��� �; ' 5 �, ! ���1 �� 2 �� 3 �� 4 �� 5 �� 6 ���� % �� � & � % �
�. � �9 3 � � �� �� : % @ $ # ��2 3 ���. � ���������� � ���� �
�������"� � � � � � � � � ( � � � � � ) � � ( � � � � � � & � � * � � � � + � � & � � "� 1 = $ ���� � � 3 � < # � ��� $ ����� 3 � ��� $ ������ $ ��� < #
B �- ��� ! @ � 3 ���� . ����� D �� �� : ���� �! �� : P J / % �� ������� . P & � ��� �������.
� � � � �� � ������� )T h e o r e t i c a l A p p r o a c h( $ ) �� 3 � �� J / % �� �������� � & ! �* ! �� �������� (�C � ��
�� � $ ��% �. �� F ' < # � 3 � ��� $ % ! ���� " ����� � < �9 $ ������ " ���3 � ��� & ' + & ' �! �� 4= % ��� " ����� �! ��.
, � " �1 : ������ � ��� �� ���8 � � �� � � � �� 3 �� � � �� � � ��
� � �� � � � �� � � �� � � � �� � �� � � � ! � �" # �� � � � � $ , � � �:
$ % ! ���� " ����� � =11 � $ ������ " ����� =3
7
∴ + �C � �! 4�� ������ =11/3 � $ ' �� �� �� �� $ ) �� 3 � � ������ 4��� � � �� �, 9
$ ������ " ����� �� 3 � ��� & ' + & ' �! �� 4= % ��� & ��� $ % ! ���� " �����. , � " �2 :
! �, 9K 5 �% . ��)4 (�9 �� 9 0 # � �� + �C ��K ! 2 �� QK �. " � )A � B � C � D (�E � ��2 � �� & ' $ ! 2 �� �) ��� ��% � �% ) � �
$ �� �� " ���N ���: � . �C ��� ��E � ������ �. ��A 4 . ��C ��� �� ��E � ������ �. ��A �� D R . ��C ��� ��E � ������ �. ��A �D . �C ��� ��E � � � ������ �. ��A
, � � � :�� L �1 �� & % �S J � $ % ! ���� " ����� $ ��� � �. S = {AB, AC, AD, BC, BD, CD}
� P �C ��� ��E � $ ������ " ����� �A �. 3 � � " ��� $ % ! ���� " �����6" ��� .
������� =
2163)A(P ==
4P 5 �% . 5 ��E � $ ���� " ��� A �� D 65 P(AUD) D)or A(P ==
$ ������ " ����� � $ % ! ���� " ����� �
8
RP ��E � $ ���� � ��� $ ��� 5 �% . D , A P (A , D) = P (A ∩ D) =
61
P 5 �% . 3 ��E � � �� $ ���� " ��� A 2
163)A( P ==∴
3 � )A( P ��E � � � ������ & . A
� �� �� � � ��� � �������� � � �����* � J / % �� ����� ���� & ' " ����= �� �� 4���� F �� 5 �% .
��% � : 1� � � � � � � � # � � � � � � � + � ( � � � � � (
������� �% � ��! �� (* ) �'SP �% �9 4# �� 0 $ ��% �� : # � " , E � ������� �� K 4 + � �� & ' 0 $ ��% �� B �- � ������ 4���.
4��% �� �% # � R 4 + � �� & ' 0 $ ��% �� B �- � ������ 4���� ��� $ ������ " ����� � �) �� & ��� & . � K 4 + � �� ���% B �� �
������� ���% B �� � 4��% � � $ % ! ���� " ����� � �) �� & ���� �� & ������� ���% �� D , . � 4��� (� 4�= �� �� : % � Q % �% % ! ��
4�= �������* � J / % �� ����� ���� .
" � * * * �
&
9
2� � � � � � � � # � � � � � , " � � � - � � � � . � + �, . �! �� $ % ! ���� " ����� �) ��� : ����� ��2 J / % �� �����' $ ���� �� : ���� $ ' ��� �� S E 2 ��E � % � (* ) �' ��I � ���� �
): ��� �� ( ����� �% % ! � * ' ��� �� �# �� �� R�� �� �� 4� �� & . �� J ��� T* ) � (* �� : % �! ������ �4/1 4��� �! � �� ��
�� U C ���� �� : % @ J / % �� ����� �� E ���� ������� �, . �� ��$ # ) ���� � $ % ! ���� " ����� D , ..
J ��� (�! , ������ ��! �� ������ �? � ����� > ���% � 5 �, !2/1 ��� $ ��� $ ������ " ����� �� ����� < # � " ������ (�! , : % �! & . � �
������ $ % ! ����)< ) % � �� ! , ( + ��% � > � & ' . �2 ��� �� : % @ 5 �, �3 �% M � < # � �! , �� ������ � � �� �. ������.
D , . �) � < # � 4# V � �����* � E W ��� < �9 $ ���� " ? 2 % �% . ��& � ��� ������� �. ����� �, . " ����= ��.
� � � � �� � �������� � )E m p i r i c a l A p p r o a c h( " * ���� J � ���� �! 2 < # � " �! $ �# � �% � �, 9N � ��! � ��
< # �� < �9 �! �� : ��� ��/ ��� �% & ��� " ����n1 & ��� " ���� �� ������� : ��� ��/ ��� �%n2 : ��� ��/ ��� �% & ��� " ���� ��
V = ��n3�; ' # � " ���� � $ ��% �! �� : � =
Nn1
10
������� : �# � " ���� � $ ��% =Nn2
V = �� : �# � " ���� � $ ��% =Nn3
K�� �� �# �E � 4�% �� D , .31 $ % ! ���� " ����� �) ��� � ��
" % �! ��# ! �N. " ��- � ��# ! ��! $ �- % � ������ �� 4�% �� D , (* ) � ��! � - � $ ���)70 % �20 % �10 %& ������ < # �.
�� ����� �! � 3 �� L i m 70.0
Nn1 =
N → ∞
: ��� < # � ��= ��� ������ �. � " �! �� $ �# � & � % � �! @ �
L i m 20.0Nn2 =
N → ∞
: ��� < # � ��= ��� ������ �. �" �! �� $ �# � & � % � ���@ �
L i m 10.0Nn3 =
N → ∞ : ��� < # � ��= ��� ������ �. �" �! �� $ �# � & � % � V = @ �
, ! $ # �� $ ��- �% � �, 9 5 �n : ��� ��/ " �� � ��! � �� �. ��=r �= �� � $ ��% �; ' �� =
n
r . �� �# �E � - $ ��% �� D , .2/1 " % �! �, 9n " �� ��# ! : % � " ��) �� �� �! �� �V = n $ ��% �� " ��- � ��# !
n
r ��2/1� �� % � : % � 3 �� ��!n �� (� $ �- $ ��% �� U �= � (� ��! 2/1
J � ��
L i m 2/1Nn1
= N → ∞
11
�% �� $ ��- & � % � ��= < # � ��= ��� ������ �. �.
� � ��������� � � ���� � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � 3 ����� " % �! �, 9A3 , A 2 , A1 .... � � �� < % ��� $ ' �% �� 3 ��� �. K�� 3
J N K D , . 3 � � ������ �; ' & ������� Q E @ � 3 ����� �� J � 3 � � $ ������ < �9
�; ' (��� �� ��! (��� 3 ����� ��! �, ; 'B � A �! 2 �� & ' ��! �' �% �� �) �� )1(
�; '= P(A) + P(B)) B �� A (P � ���� �) ���� �� B �- � ������ (�C � � �
P (AU B) = P (A) + P (B)
s A B
��� ��� ����� ������ ���� ���� ��� ���� : �� � � � � ������ � � � �� � �� ��N � � � ����
�� ���� ! � �� �" # � �� �$ % & ��� ��� � �� r ' � ( ��) � ! � ��� �� * �+ � ����� ��$
Li
m r
� �$ �# ��� ! � ��� �� ! � �� ( � * �+ � ����� ! � ��� , �� * �+ � � ����� * ��� ( ��) �
�����)1(
������� ��
12
�� �� 3 : YJ ' � < # � ��= ��� ������ �. �� % �. � & � $ ��� & '
� � �� � : < # � ��= ��� D �% �� J ' � < # � ��= ���1 �� 3 �� 5 3 �� �; ' $ ' �% �� $ ) * ) �� 3 ����� D , . ��
P (1 �� 3 �� 5 ) = P (1 ) + P (3 ) + P (5 ) =
61
61
61 ++
= 21
�� �� 4: Y����2 �� �� � < # � ��= ��� ������ �. �� ��� % �. � & � % �
���� � : < # � ��= ��� D �% �� ����2 �� �� � < # � ��= ���)1 � 1(�� ) 2 � 2 ( ��)3 � 3 ... (�� )6 � 6 ( ��% � �! ������� $ ' �% �� 3 ��� & . �
361
)6 � 6 (P+ ... + )2 � 2 (P+ )1 � 1 (P ) = ����2 �� �� � ( P =
361...
361
361
+++
= 366
13
�� % ��� ����
�� � � � � � � � � � � � � � � � � � � � � � � �) ���� & ' �% � ���2 � � � % �A � B 3 ����� �= ���� ��! )A
��B ( B �- �A- � �� �1 % � < # � � BB �) ���� B �- � �� �1 % � < # � A � B �! 2 �� �� U C � ��! ��� " - � & ' ���)2 (& �����
�O �P (A) + P (B) 3 ��# � $ ������ " ����� B �� � �) �� A (�' �C � � � ���9 " ����� B � 3 ��# � $ ������B �K� �K! �� $ K/ �* � 4K �! ��
3 ��# � $ ������ " �����A 3 ��# � $ ������ 5 # �� B $ K������ " ����� ��C �� B �- ��A � B > � $ ��� & ' : % ; ' �, ��� � (��� P (A) � (B) P > K� % �K% % ; '
s
A B
A and B
� � � � �B , A � � � � � � � � � � � � � � � � � � � � P (A) + P(B) = 1
� � � � �B , A � � � � � � � � � � � � � � � � � � � � � � � � � � P (A) = P (B) = 1/ 2
� � � ! � " � � � � � � � � # � � �A # � � � $ A � � � � � � � � � � A� A � � � � � � � % � & % � � � � � � � � �� ' � � � � � � � � � � � � � � � � � �
� % �
���)2(
� � � � � � � � �� �
14
)B � A (P [ � �� �� �, �� � ��� )B � A (P �= �% � � ��� �� ������� < # � )B �� A (P�, . � �. : )B � A (P – P (A) + P (B) )= B �� A (P
�� P (AU B) = P (A) + P(B) – P (A∩ B)
�� �� 5:
������ �. �� � " �! 2 �� Q �9 �� 9 0 # � + �C �� ��) � < �9 B � ��� �C ��� ��E � �� ��A �C ��� �� D �� �� 4 Q �9 �� 9 0 # � �� + �C ��
" �! 2 ��) �.A � B � C � D (Y " ���N ��� �� & ' ��# ) ��� ���� � :
$ % ! ���� " ����� B �� � �. � & % ��� L �1 �� S =
{ }CD BD, BC, AD, AC, ,AB P ��E � $ ������ " ����� A & . (AB, AC, AD) P ��E � $ ������ " ����� D & . (AD, BD, CD) P ��E � $ ������ " ����� A � D & . (��� (AD)
(����� ��# � �% # = � & ��� $ �% �� < # � �= �% " ������� > � � ��- 4��. P (AU D) = P (A �� D) = P (A) + P (D) – P (A∩ D) =
61
63
63 −+
15
�� �� 6: �� ����� � ��� ��� ��� ����� �� ��� ���� ��
� � � ���� �� � �� � �� ����� ������� �� �� �� ����� �� � !" ��� ���
���� � : �� F 1 %A ��� � < # � ��= ��� �) � B < # � ��= ��� �) �
�! ��� $ % �� $ ��� � �� $ - �"J �% ��" P (A) =
5240
P (B) = 5213
P (A∩B) = 5210
" ������� > � ��% �- ��� P (AU B) = P (A) + P (B) – P (A∩B)
= 5210
5213
5240
−+
= 5243
� ��������� �����
�) ���� �% � ��! �, 9 �# ������ A � B �; ' P (A∩B) = P (A) P (B)
� � � � � �� � � � �( � � ) � � * � +� , � � � � � � � � � - * � � � � �+� . , $ � / . , $ $ � � � � � 0 � 1 � � � � �� � � � � �� � 2 � .
16
�� �� 7: : ��� < # � ��= ��� ������ �. ��)3 � 3 (Y % �� � �� �� R�� & � % �
� � ���: : ��� < # � ��= ��� ������ ��3 �. % �� �� ��@ � ��� & � Q �
61
�= �� ������ 5 �, ! � : ��� < # � �3 �. % �� �� & % �) �� ��� & � Q � 61
P (A ∩ B) = P (A) P (B) =
361
61
61 =×
� � �� � � ������� Conditional Probability
$ � M � " ����� �� ��� , �E �� �� " �! 2 �� + � � ��� � ��� ��3 � � ������ $ ' ��� $ 3 � ��� A E W (�) � ��! �, 9 B - 3 � & I �� �* �9 2 % � ��- , E � �� ��� �� (���# �� ��! �� (* ) �' �
�� (�= E 2 �� ������ �. �� $ ' ��� ����� ��! : % ; ' ��� 1 # ��� & ' �� $ �# �� $ �# ��� J �2 �) 3 ����A ( & ' �* �M � . �2 �� �= � - ��! �, 9 ��� 1 # ���
) 3 ����B.( 3 ���� B �- � ������ �; ' �, ! . �A 3 ���� B �- � �2 � B < �
��= < # � 4�! � & �2 �� ��������P (A/B) 3 ���� ������ ���� A �� 3 ���� B �- �B.
17
& ����� ��) ��� � ����� & �2 �� ������� ���1 � U C �% � : �� �� 8:
< # � J ��� 0 !3 � + � �� " �! 7! + �C � " � � ��9 �� �� D � < # � �! ��! : % � �% ��� �% % � F 1 % � . 4�� < �9 �% � � �, ; '
� ���� + �C � �!A � ���� + � �� �! 4�� < �9 � B $ �! ��� " ����� �; ' & # ��� �) ��� ��# � �= �% & ��� :
P ����C � ���! �� AA P + �C � $ % �) ��� + �C � < ��@ � AB P ��� + � �� < ��@ �+ �C � $ % �) BA P ���� �� ���! �� BB
4����� $ % �) �� �) < ��@ � �! �� 4��� H I ��% �� " ���������� ����C 9 �! � �& ����� :
3 � � 4 5 � �� �
� � � � � � 5 � �� �
A
B
A
A
B
B P(A) = 7/10
P(B) = 3/10
P(A/A) = 6/9
P(B/A) = 3/9
P(A/B) = 7/9
P(B/B) = 2/9
18
& # ��! : ���� �! � 4! ��� ������� �; ' �, ! . � : + �C � $ % �) ��� + �C � < ��@ � �! �� ��! � �� ������ =���� �� ��
��! � �� �2 � + �C � $ % �) �� ��! � �� ������ & ' ���C � + �C � < ��@ � ��! �+ �C � < ��@ �.
P (A∩A) = P (A , A) = P (A) P (A/ A) =
9042
96
107
=×
) �����K� P (A∩B) = P (A) P (B/ A) =
9021
93 .
107 =
P (B∩A) = P (B) P (A/ B) = 9021
97 .
103 =
P (B∩B) = P (B) P (B/ B) = 906
92 .
103 =
& �2 �� �����* � & ����� ����� � �% � ��) ��� �, . ���% �O �� :
- �. / � ��� ��� ��0 � ��� �# �� � ��� ��B , A ���� P (B) �1 $ 2 3 � ( ��) � �
! � ���� �. / � ������ A ! � ��� * �+ � . / � B �. �� � ����� � �� ������ :
P (A/B) = P )B(P)BA( ∩
! � ���� �. / � ������ �� ( �A ! � ��� * �+ � . / � B ( ��) � � 4 ��� ������ � �) + �3 ��B , A ! � ��� ����� ��� B
19
� & �2 �� ������� ��4! ��� ������� H �% ��% �� �% % ! � D * � B) P(A P (A∩B) = P (A) P (B/ A)
= P (B) P (A/ B)
$ ��� & 1 ' �) �� �� ) ! @ $ V = �� D , . ���% �� �% % ! ��3 ��! � 3 ��� P (A∩B ∩ C) = P (A) P (B/ A) P (C/ AB)
�� �� 9:
< # � J ��� 0 !7� + �C � " �! 3 : % � " ��� + � �� " �! 3 Y+ �C � ��# ! ��! � �� ������ �. �� � ��9 �� � " �!
���� � : �! � A1 < ��@ � 4���� & ' + �C � �! < # � ��= ��� �.
A2 & % �) �� 4���� & ' + �C � �! < # � ��= ��� �. A3 �! < # � ��= ��� �.3 ��) �� 4���� & ' + �C �
��% ��# � 2 ����� �������� P (A1∩A2 ∩ A3) = P (A1) P (A2 / A1) P (A3 / A1A2)
=
85
96
107
= 720210
20
�� �� 1 0 : ��� � �� ���- � " ��� �� ������ �. �� � ��9 �� � 4�# �� ���� $
Y��2 � ��! � ���� � :
�! � A ��� < # � ��= ��� �.10 < ��@ � $ - ��� & ' B ��� < # � ��= ��� �.10$ % �) �� $ - ��� & ' P (AB) = P (A) P(B/A)
= 513 .
524
� � � � �� �� � � � � � � � ��� � �� � � � � �� � � ��� � � �
J � " ������ 4����� " ��# ��� D , . �) � �= �� $ ���% ��� ���� ��� 3 � �& ����� ��) ��� & ' $ �C ���� $ % ���� � 2 @ � $ �� :
�� ��� 1 1: & # ��! R� � $ ) * ) �% � :
R ���I � 6 ��% � �� � ' $ �2 � 2 ���% � . R ���II � 8' $ �2 � ��% � �� � 3$ �% � . R ���III � 4 ��% � �� � ' $ �2 � 1& �% � .
� �" �# �� � �5 �/ �� � ������� � 3 & � : ! ��� � ������� 4 ��� # � �� 6 �# � � � � 6 �� ��� ����' �. �� � ������� 7 � �
21
$ I ��2 � $ ��� (��2 : % � 4��� $ I ��2 � $ ��� � �E �(�C � . ������� �. ��'(P)(��� �2 �� ��! �� .
���� � : ��� � �� $ ������ $ I ��2 ��� $ # ���� ��) ��� �, . & '
)i( ) �� �� � ��E % R� � $ ) *. )ii( (��% � ��! �� ����� ��2 ��E %(A) (��� �� (B).
��� �� B ' �! � ������� & ���� $ # ���� D , . $ ����� $ % ���� � 2 �� �) ��� 2 # �:
4C �= ��� (����� $ ���� " ������ �� ��� J � 3 � � ������ ��! B ' �! � " ������������ �, . & ' . R �� ��E � (* ) �'II (��2 ��E � �)
�. (��% �
83
31 .
B �� � ��! & �� �2 < �9 J N � $ ' �% �� " ���� $ ) * ) �� : % � 3 �� 4�# ���� ������� �. $ ) * ) �� " ������ D , . " ������) ��= ��� ������ J �
& �� �2 < # � :(
I II
III
A B A B A B
3/1
3/1
3/1
6/4 6/2
8/3
8/5 4/1
4/3
22
P = 43 .
31
85.
31
64 .
31
++
=
++43
85
64
31
= 7249
2449
31 =
� � ��� � � ����� �� � ) T h e ore m'Baye s( $ ���� $ �# � " � 1 � ��# E � � �- � % = �% � �� �% C �' � �� . D 1 �
�. �� ��% �� 4�% � � ��� �� �- � % = �� �� �� (�I ��2 � " ��� : �����������@ � �� % = �� �� " ��� - $ ����� � 1 ��� D , . ��! � �� .
& ���� � � $ / % �� � ��- � E ��% B �% �� �, . �� $ # I �� < # � $ �� ] �& �2 �� �����* � (����� �. ����� �! �.
< # � + �% � F �1 �� $ �= " ������ 4��� < �9 � � $ / % � ��$ � � �� $ % � � " ���# �� .� ��! 4��V �� & ' " ���# �� ����@ � � Q
$ % �� $ ���� �� ��� 3 �� �� $ ' �C 99 � �� �� $ = E 2 �� : ��E �* E �� ��$ �? ���� �� 3 ���� �, . & C �� �* E . ��E �� < # � ���� & ��� " �������
& # ���� �������� < � � $ � ��� H I ��% < # � ��= ��� ��- � $ = E 2 ��Prio r Pro b a b il it y�� " ������� (* ) �' < # � �� $ ������ " ������ " * � < # � � ����
& # ���� �����* � $ # ) �� ���� �# ��� � R��% M � $ �! � ������ �- �� . �� % � ��� J ���� �������� < �� $ ���! � $ % � � " ���# �� + �C < # � ������� 4��
Po s t e rio r Pro b a b il it y .
23
' ��! - > I �2 �� ��@ � � � $ �� % � (* ) �' � � �% % . , && . F �1 �� D , . � � ��� �, �� $ = �E F �' :
A1 > I �2 �� ��@ � � � ��! �� 1 A2 > I �2 �� ��@ � � � ��! �� 2 A3 > I �2 �� ��@ � � � ��! �� 3 ^ ^ ^ ^ ^ ^ ^
^ $ ' �% ��� $ # ��2 $ ��� � �) �� " ����� D , . �.
$ % � � " �% �� " �� �, 9B 4��% �� �% ������ �. �/ �� D , . �� P(A4/B)�� < �� ������� �, . �; ' � . ���� E ��� �! � � � � ��- �� > �
�# ��2 �) �� �� ) ! @ ����� F ��% � ������ � � ��� �% % � �9 �' �% ��� ��' �' �% �� �# ��2 �) �� < # � $ / % ��.
A10 10
- 8 � � � �9 # ��� ��A � B �# ��� : 2 � �$ ���$ �# ��� �����/ ��0 � �� S
�D! ��� : 2 � ; 2 # �$ ! � �� ( � O ≠ P (D) ��$ P (A/ D) = D) B(P)D A(P
)D A(P+
P (B/ D) = D) B(P)D A(P)D B(P
+
S B A A
D
24
� � ��12: ��! �, 90 . 40 � ��� B �% ��# C 1 �� $ % � & ' �% E ��� �� A
B �% �� ��# C 1 ��% � �- �����B �# ) � + ��% �� �! �, 9 � 0. 30 �, �� �� �� ��# C 1 A � 0 . 40 ��# C 1 �, �� �� ��B . �� $ I ��2 � $ ��� �% �E � �, ; '
B �% �� ��# C 1 ��� ��! � �� ������ �. ��' ���� " % �! � �% E ���AY ���� � :
& . " ������� �� F 1 % � : (A) P B �% �� �C 1 �E ��� �� ������A P (B) B �% �� �C 1 �E ��� �� ������B
P (M/A) ��� ��! �� ������ B �% �� �� �2 � (* �E A �C 1 ��� �. P (M/B) B �% �� �� �2 � (* �E ��� ��! �� ������B�C 1 ��� �. P (F/A) D ��� �E ��� ��! �� ������ B �% �� �� �2 � A�C 1 ��� �. P (F/B) D ��� �E ��� ��! �� ������ B �% �� �� �2 � B�C 1 ��� �.
��C � $ % ! ���� " ����� " ������& ����� �! 2 �� � :
A
B
M P ( A M ) = P ( A ) P ( M / A ) = 104 .
107 =
10028
P(A) = 4/10
P(B ) = 6 /10
P(M /A) = 7 /10
P(F /A) = 3 /10
P(M /B ) = 6 /10
P(F /B ) = 4/10
F P ( A F ) = P ( A ) P ( F / A ) = 104 .
103 =
10012
M P ( B M ) = P ( B ) P ( M / B ) = 6 / 1 0 . 6 / 1 0 = 3 6 / 1 0 0
F P ( B F ) = P ( B ) P ( F / B ) = 106 .
104 =
10024
25
�� � ��- �� E����� P (A/F) = 24/100 12/100
12/100 (BF) P P(AF)(AF) P
+=
+
= 31
3612
=
) �% �� �% % ! ��& � ��� � � � � �� " � ���� � � D , . � ! � : ���6 ) � , $ �
���6 2 � � �
���6 � 7 � � �
�6 ��� � $ - �
� � �����
3612
3624
10012
10024
103
104
104
106
A B
1 . 0 10036 1 . 0 � ���8 �
� ��������� ������� Repeated Trials
H I��% � � < # � � �= � � � " � ���� � 4 ��� $1 ! ��� � � �, . & ' 0 % � " � �� � �� (�% � � (� � $% � � $� � � ! � < # � 4 � �� �� �! � & �� � $1 # �E�� �
$1 # �E� � � / " � � �� � � / � � 0 1 % " � �. +6�� :6 9 � � � - * � 5 � � � � 2 � �8 � 9 $ �� � ���
� % $� � - + �� � 9 % � (* ) �N �% % ? ' � �= : �� � �� / � �% � � �% ! � � � < # � � = � % � ' � � �= : �� � �� / " � � � � $1 # �E� H I��% 5 �% . �? � / � * %
)N$���! 1 = � � �= ( ��)1P N$���! � � �� � �� � �= ( �� )2PN$���! �� �� � �= ( �� ... ��) � � �= 1 =N���! $ ( 5 �% . �� J �
26
1+ N $% ! �� $� �� . " � �� � � �; ' " � � 0 �E � % � � $� � - �% � � � �� (* ) �'& . $% ! ��� �: 5 � �= � $���! 1 = 4 �= � � � �� $���! 3 �= � ������! 2 �= � 3 $���! 1� �= � 4 $���! � �= 1 = � 5 $���!
" � �� " � $% ! ��� � " � �� � � B �� � �� J � � � ���� � � " �, $# � ���� � � ! ��� � 4 � �� � & ' " � ���� � � D , . � ) � 4 ��� �
� � � � & I�% ) & � ���� � � ��% �� � � � E��% " ��) � �.
� � � � �� � � �� � � ����� � � � � �� �� L a wB i n o m i l P r o b a b i l i t
& �O � J % ��% �� � � �, . < � 9 � = ��� � $1 ! U C ��� � : � � �� $� � - < # � � �= � � �� / � ���� � =½
< # � � �= � � �� / � ���� �3 �. " � � 3* ) � � �� $� � - $� � �� (�� � > � - ½ . ½ . ½ = (½)3
�� / � ���� � ��! " � � 0 �E � � �� $� � - �� �� % > � - 0 �E �% � �, ; ' ��� � � � � < # � �. �� / � � : �1 % " -�� � & ' & % � �� % � > � - 3* ) < # � � �= � �
J � ��-��� � =(½)2 .���� � ��! 5 � , �� > � - 3* ) < # � � �= � � �� / � " -�� � 0 1 % & ' (�� � ��-��� � ��� � � � � < # � �. �� / � �� =
(½)3 (½)2
27
� �� < � �@ � " �� 3* ) � � & ' � �= � � �� / $ �% & . D , . �! � J � & ' � �= � � �� / � ���� � 4 ��� % �% % ! � � ��-��� � �� �� � & ' �. �� /
��� � - J � & ' �. �� / � �� > � - 3* ) ��� > � � � � 4 � � �� / % � � F V � �� / � ���� � �� J � � ���� � � ���� � � �� �! � 4 �# � �� � � ���� � � �� U C ��� � � � � > � - 3* ) < # � �. �� / � ���� � �� �! � > � - 3* ) J � < # � � �= � � .
" ���� �� � � 4 �� % � � ' ���� � � E��% 4 �# � �� � � ���� � � < # � � �= � # � �� - 0 �E �� �� % �! � �! � & �� � > � - 3* ) �� $��� � � ! ��! �� 3� � >
��! �� 3� � > � - 0 �E �� �� % �! � �! � & �� � � ' ���� � � 4 �� % J � � � � ' J � > � - 3* ) �� $� ' �� � !53⊂ (�� � = ��! D * �� D �% ��� J , � � � ���� � ��
! �� " ��) � ���� � � J � 3* ) � � � ' ���� � D , . �� $��� � �� : � ' �� � ! � & ' < # � � = � % ' 4 ��� �� � � ���� � � & ' � ' ���� � � 4 C % 5 � , � � $��� � � !
53⊂ (½)3 (½)2
= 1 0 (½)5 =
3210
�� 3 �� B �-� �� � �% $% � � $� � $� �� & ' : % � � �� % ' ��� % �� > � ��% �O ��B �-� � ���� � ��! � : ��-� � � �� 3 �� � � �, . (P) : ��-� � � � ���� �� (1-P)
$� �� � D , . �% ! �(n) & ' 3 �� � � �, . B �-� � ���� � �; ' � � (r ) 3� � � )0 > r> n ( ��! =
nr⊂ p r (1 -p )n-r
& ' � E�� �� ' �% ! , ��! � � � � � & I�% ) & � ���� � � ��% �� � � ���� � � �� �, . �# � ���� � " � �� � � �� ��-� � � �� � �= � � B �-� � ���� �' � " ��) � � � ���� � � " �, $
$� � ! & ' " ��)= 21
28
��� )13:( " � � 0 �E � % $� � - & � �. � � ���� � �% � �) � < � 9 � �� � �� �� / � $= �E� � " � ���� � � �; ' � � �� � � > � - 0 �E & � ��
��% �� � � �, . �� E���� �� ���� �! � : �� �& # ��! : � �= � � �� / � ���� �5 " � � =
( ) ( ) ( )321
21
21 2
1 5055
5 ==⊂
� �= � � �� / � ���� �4 " � � = ( ) ( ) ( )
325
21 5 2
1 21
51454 ==⊂
� �= � � �� / � ���� �3 " � � = ( ) ( ) ( )
3210
21 01 2
1 21
52353 ==⊂
�� � � �= � � �� / � ���� � = ( ) ( ) ( )
3210
21 01 2
1 21
53252 ==⊂
= � � �� / � ���� � � � �� � � � � = ( ) ( ) ( )
325
21 5 2
1 21
54151 ==⊂
� �= � � �� / � � � ���� � = ( ) ( ) ( )
321
21 1 2
1 21
55050 ==⊂
29
�� / � * %$� ����� � � � � � & . H I��% � � D , .� � � � & I�% ) � � �� � 5 �! 1 � & ' ( )5212
1 +
� 8 � �� � ! ��� � 4 � �� � & ' 3 �� � � B �-� " � � � ��! ��� �. � & ��n - �� 1 = �� � � , E? �1 �� 2 �� ... ��n �; ' $' �% ��� $# ��2 " � �� & . �
1)n(P...)2(P)1(P)O(P =++++
� � ��14 : $� �� � �� / � $% ! ��� � " � �� � � = � �3 & � % � �� �� ���� � 4 �� � �
% � � �3" � � . ����� : $% ! ��� � " � �� � �
P : �� � �� / � � 3����� P D �� / � � �� � � P �� � D �� / P " � � 3* ) D �� / P : �� � �� / � � � ���� � 3 (����� =
216125
65
61
3030 =
⊂
P � � �� � � D �� / � ���� � =
21675
3625
61 3
65
61
2131 =
=
⊂
3 0
P�� � D �� / � ���� � =
21615
65
61
1232 =
⊂
P " � � 3* ) D �� / � ���� � =
2161
65
61
0333 =
⊂
" � ���� � � B �� � �� / � * % = $' �% ��� $# ��2 " � �� � � D , . �@ 5 � , � � �� � � � � & I�% ) � � �� � 5 �! 1 � & ' $� ����� � � � � � & . H I��% � � D , . �� / � * % ��!
3
65
61
+
+�� � �� : � � - * � � � � 5 � � � � 2 � �8 � �� & ' " � ���� � � �% � � ���� � + � � � & ' " � � � ! ��� � 4 � �� � $�
+ � � � �, . & ' � " ��) � � � ���� � � " �, $# � ���� � " � �� � � & ' J � � � / � � 0 1 % B �-� � ���� � �� ' V � & �� �� � ! ��� � 4 � �� � $� �� & ' " � ���� � � 0 % � ��% �� � � � E��% " � �� � � D , . � ) � & ' � Q E� < � 9 $� � �� 3 �� � �
J ��� ��� � � .
� ������ � ��� �� � � � � � � � � Hypregeometric Low
& � ��� � � �) �� �� ��% �� � � �, . U C �% �� �% % ! � : � � ��15 :
: # E� � 0 ! �% , E� ��N & . ��# � � � � D � �� � > � �� $# ) ���� � ! :n1 � + �C � � ! n2 � + � �� � ! n3 �. � " � ! �% �� �� + � �� � ! R
3 1
��! �� � ���� � 4 ��� 4 �# � �� �� � ��9 �� � $��� ��� � " � ! � � �� ��r1 � + �C � � ! r2 � + � �� � ! r3+ � �� � ! .
����� : ��E� ��R �. � � � � 0 ! � � �� � ! ⊂
NR " � �� � � � �. �
4 � �� 3 �� � � �� � $% ! ��� �r1 K � � + �C �� � " � ! � � �� + �C � � ! n1 �, . � �. � � � � ��⊂ 1
1
nr
� 4 � �r2 K � � + � �� � � " � ! � � �� + � �� � ! n2 �. � � � � �� �, . �⊂ 2
2
nr
� 4 � �r3 + � ��� � " � ! � � �� + � �� � ! n3 �. � � � � �� �, . �⊂ 3
3
nr
�. � � � � �� 3 �� � � �� J � ( )( )( ) 3
3
2
2
1
1
nr
nr
nr ⊂⊂⊂
���� � " � �� � � � �. �, . � $� � ���� � � ��! & � ��� ��� =
⊂
⊂⊂⊂NR
nr
nr
nr
3
3
2
2
1
1
�K K3 R = r1 + r2 + r3
= n1 + n2 + n3 ����� ����� ��� �� ������:
: # E� � 0 ! �% � ��! �, 9N �� % � $# ) ���� � ! n1 � �@ � ��# � � �� � ! �n2 � & % �) � � ��# � � �� � ! ... �nz��# � � �� � ! Z � � ��9 �� �% �� �R
�������� ���� �
�� � ���� ���� �
3 2
$��� ��� � " � ! � � �� �� ��! �� � ���� � �; ' � !r1 ��# � � �� � ! � � �@ �r2 � & % �) � � ��# � � �� ... �rz ��# � � �� z �. :
⊂
⊂⊂⊂NR
nr
nr
nr
z
z
2
2
1
1...
J ��� ��� � � & � ���� � � ��% �� � �� < � � $V = � � D , . � � � ��16:
�� ' $��-10 � 4 * � 5� " , E� " ��� �� �� � � $I��2 � : % 20 %� ���� � �. ��' :
(i $�� �� � ��� �� �� ��! �� ��. (ii" ��� �� < # � J ��� � �� . ����� : (i $�� �� � ��� �� � ���� � ��! J ��� ��� � � ��% �� � � � �� �� =
455225
153
51
102
=
⊂
⊂⊂
(ii # � $�� �� < # � J ��� � �� & % � " ��� �� < # � J ��� � �� � ���� � < 4 * � � � �� (�� � ��% �! �� � ���� � S -�% � �� J ��� �, . � � -@ �
455335
1 153
50
103 =−=⊂
⊂⊂
" ��� �� 3* ) �� ���� �� �� $�� �� < # � J ��� � �� � ���� � �. �� 455335
153
100
53
101
52
102
51 =
++
⊂
⊂⊂⊂⊂⊂⊂
3 3
� ����� �� � � �� � � �� � � � �� �� � � � � �� Expectation
% � F V � � $# �� �. & C � � � > -��� � " � ���� � � $ / % � � ��� ��� �� �% C �' � �, ; ' � $� ��� � � I��P � ��V � & ' S E2 [ � % � ���� � < � 9 �� �
� ��X 4 C � = �� �; ' : � � % % � : # � � = � J , � � a # ��� � $�- < � 9 �� � P.X� ��V �� � D , . �� : # �� �� S E2 � � > -��� < �� .
∴ > -��� � =��. � � � � �× 4 �! � � � ���� � �� � �� > -��# � �� % �E
E = P.X a # ��� � ��! �, 9 � �, .χ � � > ' � � � � �� n " % �! � ���� � �� i & .
> -��# � $� �� � � $�� � � �; ' � I�1 � � � � �= n
n
)i1(X . P)i1( X.P
+=+ −
� � ��17: 4 �* � � U � ��. � � 4 �� � � �� $�� � & '5% �% $- � 4 � � �, 9
4 � # � � � � �� $��� � �� � � ��)52$- � ( B �% � � �� $��� ��� � $- �� � " % �! �"& ���� "Y 4 � � $# �� � ! + �� � : � ' 4 J , � � � �� � � ��) � � �. ��'
����� : U � � � � ���� � = $- � 4 � � � ���� �"& ���� = "4
1 � � � $�� � � ���� � � �, . �% � C �, ; ' �. � > -��� � $�- < # � � �= � � � �% % ! �� ��I
: �� � � ! � � �� � � ��) � � E = P.X. = 1.25 (5) 41 =
�% � � > � � �% J ��� : �� � � ! � � �� � � � �� � �� J �.
3 4
� � ��18 : & C � � � > -��� � a # � �!)��. � � � � � J � ( 4 �� : � ' �� / �% �� �
��. 4 �! & ' 4 � D - 100 4 �! � � � ���� � ��! �, 9 �% 0. 4 0Y ����� : & C � � � > -��� �==×== 40 100
10040 .X.PE
� � ��19:
� � > ' � � � � ��� � ���� � � �) �� � & ' �! , �� � ��. � � $�- �� F 1 ��5 � I�1 � � � � ��! � " ��% �3 . 5 %& C � � � > -��# � $� �� � � $�� � � �; '=
n)i1(X.P E
+=
5(1.035)40 =
1.1876862
40 =
= 3 3 .6 7 8 9 = 3 3 .7 < � 9 ��. � � $�- > ' � " � � � & �� � $� � � " ���� �, 9 : % � �, . < % � ��
��* � � �� $�% �� � " �! � �2 � � � = � �� ��I�1 � ��� ! -� �� % � 3 3 . 7 � �% " ��% � 0 �E � �� � " �! � �2 � � D , . $# � �; ' ) $�! � $�% � � I�' > -���
3 . 5 ( % & . � ��. � � $�- a # �� > ' � $' �! ��! �100 � � �I�' � ! � �% 5 " ��% �.
3 5
� �������� � �� ����� ������� ������ : � � ��V �� � �� J , � � a # ��� � & . & C � � � > -��� � + � 2 $�- " % �! ���
& ' a # �� 4 �! � $� ��� � � ! & C � � � > -��� � � ! ' �; ' ��. � � $�- �. > -��� $�- > ' � � � �� ��? �� � " �! 2 �� , 9 ��� � � < # � ��? �� � � ! ' 0 ��� ��� � D , .
��. � �): # � ��N �� � a # ��� � ( ���* �)�% �? �� � ( $�� � � > ' � ����� �, � ��� � ��? �� � F V � �� � & C � � � > -��# � $� �� � �.
� ���2 0: & . �� $�! �� � � I�1 � � 0 ��� < # � & C � � � > -��# � $� �� � � $�� � � � � ��3 . 5 % $! 2 � � 2 � � � �� & ' S E2 �. � & �� � (��% �
a # �� : � > ' � �? � " � � � ��? �2000Y �� � @ � �� a # � �, 9 �% ����� : �� �� ���� �� �. , E� �! � ��� � � - < # � + �� �� � � ���� � $�- ��
� � ��� � � � �� ���� � � � & �"��� # � $! �@ � � �E� � � �� " � �� & . �7 � 8 � 9 & ' " � �� & �� � $� ��� � � �� � � �� ��� �! ��186 7 & . � �O � < �� (� � �� � � �� � � � �� � �
�. � $ / % $� ' � $���� �100000S E2 . ��� � � - < # � + �� �� � � ���� � =
xnxl
lP +=
3� 1x �� � � % � ��� � � - < # � �-��� � � & .x 1x + n �� � � % � ��� � � - < # � �-��� � � & . x + n
�- & ! �@ � ��� � � � �� ��7 0.84314
9263778106
llP2040 ===
3 6
& . & C � � � > -��# � $� �� � � $�� � � n)i1(
X.P+
1.98978771686.28 )035.1(
2000 84314.020 =
×= = 8 4 7 .4 6 7
, . < % � �� < # � �% �N �� � �� � 2 � � � �� & ' S E2 � ! < # � �� � : ��- ��% � �� �-� �! � �2 � ��? �� � $! 2 < � 9 > ' �� �� ���84 7. 4 6 7 �%
a # �� : � > ' � �? � �. � � � � ��� �2000 �� � @ � �� a # � �, 9 �% . �@ 5 � , � � �� � ��-@ � �� " �! � �2 � � D , . �. ��) ���� $! 2 � �3 . 5 % � �� (��% �20
a # �� > ' � & 1 ! � ���� �� � ��! � : % ; ' $% �2000 (�� � / �� � ! � �% �� � @ � �� < �� �! �2 �� � ��.
� � �� � � �� � � � ���� �� ������ ���� � 4 ��� 0 ��� < # � ��� � " � ��) ��� � ��E� & ' > -��� � � ��- � �� � �
��� � [ � � ��� � 5 � , � � �) $� -���� � � �E # � $� �� � � �� � � �� � �) ���� � a � �H ��% # � > -��� � 4 ��� � $� -���� � � �E # � $� �� � � �� � � B �� �.
� � ��2 1 : a # �� ��) ��� & ' S E2 �1000 > �2 �� � � � & ' �% �� 3� ���% � D - (� ��) ��� (� �� �� � � & �� � $��% = � � & . ��% �� � �� � ! & ' � ��E� � �� U � � � " � ���� �
& # ��! : $% �� �< � �@ � 1P �. � � J � < # � � �= � � � � � � ���� � ½
2P D - � � < # � � �= � � � � ���� �)1050 ( �. �% ½ $% �) � � $% �� � 1P �. � � J � < # � � �= � � � � � � ���� �
52
3 7
2P D - � � < # � � �= � � � � ���� �)1102.5 ( �% �.
53
�� �% �# � �, ; ' �. $� ��� � � ���@ � & ' : � � ��� �� � � I�1 � � � � � 5 % � � ' S E2 � � �, . U = % ��Y � �� ��) ��� � + �
����� : � - , �E�� � 9 � � �� ��) ��� � + � 9 � ��I ��� % �� (� �� �% # � :& # ��! �% �� $# ! 2 �� � � ) ��� :
% 5 � , � ��� � � 4 ��� � ��� 4 �� � : # � � �= � � � > -���� � � E # � $� �� � � �
& # ��! 5 � , � �) ���� � a # ��� � $�- �� % � [ � % �) ��� " � ���� � � � # �E� : 1. 0 – 10 0 0 = - 10 0 0 2. 1000)05.1(
5.11022 − = 0
3 . 100005.1
1050− = 0
4 . 100005.11050
)05.1(5.11022 −+ = 10 0 0
10 0 0
0 0
1102. 5
1 05 0 0
1102. 5
2/1
2/1
5/2
5/3
5/2
5/3
3 8
�& # ��! 5 � , � " � ���� � � 4 ��� � ��� % 5 � , � � :
102
52 .
21 ==� �@ � � ���� � �
103
53 .
21 ==& % �) � � � ���� � �
102
52 .
21 ==3� �) � � � ���� � �
103
35 .
21 ==> �� � � � ���� � �
E � � � # �E�� $� �� � � �� � � 4 ��� � �% �- �� � ���� � � � " � ���� � �� $� -��' �� � $# ��� �� �& # ��! 5 � , � > -���� � � E � � J � > -��� � 4 ��� �% % ! � : %:
�� : # � � �= � � � > -���� � � E # � > -��� � $�- �� ���� � �� �� :100 $ � % � � � ��) ��� � + � ; � S E2 � � �, . U = % % �% % ? ' �, 9.
10 0 0
0 1102.5
1 050
0
1102.5
������ � � � � �� � �� � � � �� � � � �� � � 200� 10/2 1000�
0 10/3 0 0 10/2 0
300 10/3 1000
100
2/1
5/2
5/3
2/1
5/2
5/3
0
3 9
� � � � � � � � � � � � � � � � � � � � �
1. 4�#�� ���� $��� � �� ���-� "���)52$-� ( ��� 3 ��
$ %�) �� 4�� � �- < ��@ � $-��� .$ �O � "� ����� � 4��� : �. ��@ � ��#�� �� ���������� ���-��� ��! � ��
41
4. � �� � ! 2 �� ���������� ���-��� ��! � ��)J �% ( 161
R. � ! 2 �� 0 1 % �� ���������� ���-��� ��! � �� 41
2. � �� ���� S�E 2 @ � �� R��� � $) * ) �% � . (� ' R�� � ! �� �%�E �
� ����� �. ��' � �� �� � (� ���: �. ��� 0 % �� ����E ��� $) * ) �� �' @ � ��! ��
41
4 .� ���� � # ��%�! ��
83
3. � ����� 4���� � $%! ���� H I ��%�� = �� "�� 0 �E �� % $�� - " ��� %� � !.
4. �� > � - $) * ) � � ��� $ � & ' �= $) * ) �� / � ����� �. �� Y �� %��
81
40
5. � = � �, ; ' �� % $�� - & �� SE 2 ��� > C � � �= < #� $���! < #� � = � �, 9 ��� �� %= & ' + � �� "�! 3 * ) > C : %? '
+ �C � � ! � + � �� � �! . $ #���� D, . SE 2 �� ! �, ; 'n �) � � �� %= �� �� � ! 4�� . + �C � � ! �� D, . ��! � �� � ����� 4���'
61
6. : � � �@ � ��� !5"�! 3 * ) �� %� $#) ���� "�! ���! � + ��� : � & %�) ��� ����C �73 * ) �� %� "�! "�! "�! > ��� + ���
+ �C � . ��� %� � ��� ��! � �� � ����� �. ��' 0 ! � ! �� � ! "���Y + �C � � -@ � < #� )35/26(
7. < #� J ��� 0 !3 � + �C � "�! 6� + ��� "�! 3 + � �� "�! "��� ��� � � ��3 ��9 �� �� (� I ��2 � "�! � ����� �. ��' � �
Y ��� ��� �� ��! �� �� %� )10/1( 8. : � 0 !12 �� %� $#) ���� � ! 5� + �C � "�! 3 � + ��� "�! 4
"��� + � �� "�!3$ �O � "� ����� � 4��� 0 ! �� �� "�! : �. � ������ � + ��� "�! �� / )55/21( 4 .+ ��� � ��� � ! �� / � ����� )55/27( R. ��� -@ � < #� + ��� � ��� � ! ��! � �� � ��� )55/34( . ��� ��� �� �� #! "�! �� ��! � �� � ����� )44/3( D. ��� ��� �� ) ! � �� ���! 5 �%. 0 � : %� � ����� )11/3(
41
9. "* 1 ��� �� & ' > �10 0 $-�� �#� ��� �% ��� $-�� � �. - � � I � "= = E � � � �����50 $-�� �#� �% ��! � � ' � � I �1 ��
Y � �� � �� $-�� ��� ��
10. : � 0 !10 0 �� %� ��#�� � � $#) ���� � ! 45 � + �C � � ! 30 � ! � + ���25 0 ! �� < �9 " ��� �� %�� � �� � ! "��� + � �� � !
� ! "��� �) 0 ! �� < �9 " ��� �� %�� � �� $ %�) � ! "��� �)"�! �� ����� ��! � �� � ����� �. �� $) ��) < #� $������� 3 * ) ��
Y ���� � ��� � F �� 4 ���� )0 . 0 3375(
11. 4�#�� ���� $��� � �� $-� "��� )52$-� ( " �� �)$ %�) $-� "���� .$ ����� "� ����� � 4��� :
�. �� $-��� ��! � ��< ��@ � $-��� ��� 0 1 % �� $ %�) )2/1( 4. � $-��� 0 1 % & . $ %�) �� $-��� ��! � ��< ��@ )52/1( R. ��� 0 1 % � �-��� �� � ! � ��� �� )169/10( . � �= �� 0 1 % �� ��� 0 1 % � �-��� �� � ! � ��� �� )13/1( D .� �= �� 0 1 % � �-��� �� � ! � ��� �� )169/3(
12. �� % $�� - " � 3 $ ����� "��
�. � � ����� �. ���� & ' ��� ����� ����= 3 �� �+ �) * ) �� "� � )8/5( 4. � � ����� �. �� & ' ������! �� ��� ����� ����= 3 �� �
3 * ) �� "� ��� )4/1(
42
13. : #�� �� 0 � � E ? � �� � ����� ��! �, 9 32 � ���� < �9 4. , �, 9
� (� 2 ��4/1 � $#' ����� 4. , �, 9 6/1 : �� �� 4. , �, 9 . �. ��'E ? � �� � ����� : �# �� ��E � �, 9 �� @ � �� [ ��= & ' : #�� �� Y $ I ��2 � $� � � )36/13(
14. < #� J ��� � �@ � � -� %= �% � ��! �, 9 5 � $� �� � �%� 8 � �%�
< #� J ��� & %�) ��� $� #�6� $� �� � �%� 11$� #� . � ����� �. ��' (�� �� � -� %= �� �� �� (� I ��2 � 4������ � %��� ��! �� Y
)0 . 36878(
15. � �! �� �� � : #I �� � �E � & . $� �� 4�� �� # ��� � �1 � � $��� � � ��� � �#��� 0 % �� . 4��� �) $� ��� D, � � & % ��� L �1 �� 4�! �
$ ����� "� ����� � : �. � ��� "%� $#I ���� %� )4/1( 4. � �%� �� ) ! � $#I ���� %� )16/5( R . $#I ���� %�3� "�%� ) ! @ � < # )16/15(
16. � �= -�%� < �9 + �%��� "�! 2 Q �9 "� � � B , A �� � ����� ��! �, ; '
$= -�%��� < #� � = ��A �. 0 . 6 $= -�%��� < #� � = �� �� � ������ B �. 0 . 3 �. (��� � �= -�%��� < #� � = �� �� � ������ 0 . 1 �. ��'
� ����� : P $= -�%��� < #� $! 2 �� � = �� �� A $= -�%��� �� B )0 . 8(
43
17. B B , A & % ��� L �1 �� & ' � ) �� S 3 �
31 (A/B) P 2
1 (B/A) P 51 )AB( P ===
�� KK (A) P , (B) P 52 , 5
3
18. "�%! ���� "%�! �, 9 $ I ��� ! �� U ��= ��� > %�= � �� & ' M1 � M2 � M3 4 ���� < #� > %= � 0 . 30 � 0 . 30 � 0 . 40 R��%M � B �� � ��
��! �, 9 �1 � 3 � 2$I ���� 4 ���� < #� 3 * ) �� "�% ! ���� R��%9 �� 4 �� R��%9 �. . : %� � �� @ � �� R��%9 �� (� I ��2 � ����= � 4��
$% ! ���� > %= �� [ ��= ��� �, . ��! �� � ����� �. ��' 4 ��M1 Y > %= ��M2 > %= �� YM3 Y)20/3 � 20/9 � 20/8. (
19. �%��� �, 95 4�#�� ���� $��� � �� "�-� )52$-� ( � ��9 �� �
�� � � ��� $-� < #� $��E �� ���@ � D, . + ����� � ����� �. ��' � Y 4 �2 � �=)10 829/3243.(
20. $% ! ���� 1 H �%� 60 % $% ! ����� $' � �� "� �� ��� > %= � R��%9 ��2
& -���� H �%� .5 % $% ! ���� R��%9 ��1 ��% � 4 �� 2 % R��%9 �� $% ! ����24 �� .�� � ����� �. ��' & ' "�%= - $� �� $ �� � ��! � $% ! ����1 Y )38/30(
44
� � � � � � � � � � � � � � � � � � � � � � � � � �
1. i)
41
5226 .
5226
=
ii) 161
5213 .
5213
=
ii) 41
5213 .
5213
5213 .
5213
5213 .
5213
5213 .
5213 =
+
+
+
2. �. $%! ���� "� ���� � =23���� & ' ��! & . � � i( � ��� 0 % �� $) * ) �� < #� � �= ��� � ����� J �3 � �
��3+ ��% . �K��� �K�
41 8
1 81
=+
��
41)2/1 . 2/1 . 2/1()2/1 . 2/1 . 2/1( =+
45
ii ( � ���� � # < #� � �= ��� � ����� ���� �� = =
83
�� =
83
8C32 =
�
� � � �
�
�
�
�
� � � �
�
� �
�
�������� ���� � � ���� ����
�
46
3. $%! ���� "� ���� 32 = 25
� �= �� "� ����� = $���! �� "� �����=
1/32 32
5/32 32
10/32 32
10/32 32
5/32 32
1/32 32
CCCCCC
50
51
52
53
54
55
=÷
=÷
=÷
=÷
=÷
=÷
4.
81 .
21 .
21 .
21 ==) S S S( p
5. $��#�� �� $ ��%�� "�! �� � =3nC : %@ >3 � ! > � "�! $ �
� �= < #� � �= ��� � ����� = $���! < #� � �= ��� � ����� =2/1 ∴C & ��� "���� � �� ' >3 + � �� "�! = "���� �
� ' > C & ��� J ��� � + �C � � ! � � �� �� � �! �2n
∴ + �C ��� "�! �� � = 2
n 1 2n
=× + � ���� "�! �� �= 2
5n 2 2n 3 2
n=×+×
47
+ �C � $������� � ! �� ��! � �� � ����� =
61
3n1
2n
3n 2n
=×=
÷=
6. & . � �= ��� $%! ���� "� ���� : + �C � + ��� + ��� + �C � + �C � + �C � + ��� + ���
! ��! � �� � ����� + �C � � -@ � < #� D = ���! �� ��! � �� � ����� S-�% ��� ������.
3526
359 -1
73 .
53 1 ==
−
�� � � � � � � �! � �� � ����� + �C � � -@ � < #� D! � = � �����+ ��� $ %�) ��� + �C � < ��@ �
�� + �C � $ %�) ��� + ��� < ��@ � �� + �C � $ %�) ��� + �C � < ��@ �
3526
74 .
52
74 .
53
73 .
52
=
+
+
=
�������� ���� � � � � � ���� ���� � � �
48
7. 4�#� ��� � ����� � = �� + �C � "�! 3 * ) �� ��! � �� � �����+ � �� 3 * ) �� �� + ��� 3 * ) ��
101
أو
101
101 .
112 .
123
104 .
115 .
126
101 .
112 .
123 P
CCCC
123
33
63
33 =++=
=
+
+
=
8. $%! ���� "� ����
220 C123 =
i) P )� � � � � � � � � � � � �( = CCC
123
30
93
= 5521 220
84=
ii) P )+ ��� � ��� � !( = CCC
123
92
31
= 5527
iii) P )+ ��� � -@ � < #� � ��� � !( =
=
5534
��
CCC
CCC
CCC
123
90
33
123
91
32
123
92
31 ++
49
= 1 -P )+ ��� � ! �� / � �( = 1 -
5521
= 5534
iv) P )����#�� 0 1 % �� �� #! "�! ( =
CCCC
123
33
43
53 ++
= 443
v) P )��#�� 0 1 % �� � �! � � � �( = 113
. .
CCCC
123
31
41
51 =
9. ���� = � ����� � × � � I � �� $� -
50 × 1001 =
21 =
∴ �% � = % ��! �� 4 � ���� ����. ∴� �� � $-�� ��� �� .
10. 4�#� ��� � ����� �
10030 .
10025 .
10045 P =
= 0. 0337 5
5 0
11. i ( < ��@ � $-��� & ' ��#�� < #� � �= ��� � ����� 0 1 %
=2/1 ���� ��� � � ' � %�� 5 �%. �@
ii( % < ��@ � $-��� � ����� 0 1 =521
4�#� ��� � ����� � = � �-��� ��! � �� � ����� 1 �� 2 �� ... ��10
16910
1310
524 .
524 10
. . . 524 .
524
524 .
524
3
=
=
=
+
+
=
iii ( 131
524 .
524 13 =
=
12. i . $%! ���� "� ���� �8
= S
$ ������ "� ���� �5 ∴ 4�#� ��� � ����� � =
85
ii. 82 P =
41 =
� � � �� � � � � � � � � � � � � � � � � � � � � � � � � �
5 1
13. $) * ) �� "�� �E � � �� � ! � ����� =)3/1( � �� E ? � � (� 2 �� 4�. , �� � ����� �. 4�#� ��� � ����� � $#' ����� 4�. , �
E ? � � : �� �� 4�. , �� �� E ? � � P ) E ? ���( =
61
31
41
31
32
31
+
+
=
3613
14. �� %= J � � �E � � ����� =2/1 4 ��� � �@ � �� %= �� �� 4�� - ��! �� � ����� =
135
21×
�� %= �� �� 4�� - ��! �� � ����� 4 ��� & %�) �� =176
21×
P )4 ��( =
+
176
21
135
21
= 0. 36 8 7 8
15. S = 24 = 1 6
i) P )� ��� "%�( = 16C41
ii) P )� �%� �� ) ! �( = 16 CC 4
443 +
iii) P )) ! @ � < #� "�%� 3 * )( = 16C 144
−
= 1615
� ��� & I �%) ��%�- � �� �� �� = 16
CCCC 43
42
41
40 +++
5 2
16. P (A �� B) = P(A) + P(B) - P(AB)
= 0. 6 + 0. 3 – 0. 1 = 0. 8
17. P(AB) = P(A) P(B/A) = P(B) P(A/B) ∴ P(A) = 5
22/15/1
)A/B(P)AB(P
==
P(B) = 53
3/15/1
)B/A(P)AB(P
==
18. �! ��
P(D) =�� �� [ ��= ��� ��! �� � ����� P(M1) = > %= �� [ ��= ��� ��! �� � �����M1 P(M2) = > %= �� [ ��= ��� ��! �� � �����M2
P(M3) = > %= �� [ ��= ��� ��! �� � �����M3 "� ����� � & � �� $�? ���� "�%� � $ �O �
P(M1) = 0. 3 P(D/M1) = 0. 01 P(M2) = 0. 3 P(D/M2) = 0. 03 P(M3) = 0. 4 P(D/M3) = 0. 02
5 3
$ ����� "� ����� � 4��% �O � �%� P(M1D) = P(M1) P(D/M1) = (0. 3) (0. 01 ) = 0. 003
P(M2D) = P(M2) P(D/M2) = (0. 3) (0. 03) = 0. 009 P(M3D) = P(M3) P(D/M3) = (0. 4) (0. 02) = 0. 008
���� � D, .� %� � �� # ) �� �! � (�� � "� �. �� E ���� � � $ / %
203
0.0080.0090.0030.003
)DM(P)DM(P)DM(P)DP(M )D/M(P
321
11
=++
=
++=
)DP(M )D(M P )D(M P
)D(N P )D/M(P321
22
++=
= 209 020.0
009.0=
208 0.020
0.008 )D/M(P 3 ==
5 4
� � � ��� $� ����� "� ����� � S E #� �! � �& ���� : ������ � � �� � � � �� � � � � � �� � � � �� � � � � �� � � � �� � � �� �� � � � �� �
M1 0 . 3 0 0 . 0 1 0 . 0 0 3 3 / 2 0 M2 0 . 3 0 0 . 0 3 0 . 0 0 9 9 / 2 0 M3 0 . 4 0 0 . 0 2 0 . 0 0 8 8 / 2 0
������� 1.00 0.02 0 1.00
19. Q ��� � �� �� ��%�� �� �� E ����
c
cc525
44
41
48
) = ��� 4 �2 (P
108293243 =
��� �� & ' �� C �� �� � #�E � 3 ���� �, . B �-�� $ ' �%�� "� �� 0 �E 5 �%. �� ' � / & ��� < ��@ � $����� , E ? %� � 4 �2 �� $-� 4�� 4 ��
� ����� � ��! $����� D, . & ' 4�� � �� �� 4 �2 �� = 541454243
4845
4946
5047
5148
524
=××××
�' $#) ���� 0 �E �� "� ���� �� 3 �� �. 4�#� ��� � ����� � �
108293243
541453243 5
=
=×
5 5
20.
5 0.789 3830 0.038
0.03 0.0080.03
0.03 )DM(P)DM(P
)DM(P)D/P(M221
11
=
==
+=
+=
M1
M2
D
N D
N
P( M) = 0 . 6
P( M2 ) = 0 . 4
P( D / M1) = 0 . 0 5
P( D / M2) = 0 . 0 2
0.03 0.05 0.60 )M/D(P )M(P)DM(P 111
=×=
=
0.008 0.02 0.40 )M/D(P )M(P)DM(P 222
=×=
=
5 6
��������� ��������
57
������� ����� � ��� � � � � � �� � � � ��� � � �� � � �� � �� � � � ��
RANDOM V ARI AB L E S AND P ROB AB I L I T Y DI S T RI B U T I ONS
& � � � % � � � . � � , � � ) � 4 � � � � � � �� � � � = 1 � � & ' �% � � �
4 � # � � � � � � � � $ - � 4 � � � � D ! 4 � � � � � % � � $ � � - ... H I �� % � � � � � c � � % � � $ � �� & ' �� ! " % �! $ � - � $ I � � 2 � 4 � � � �� # � � � = � � � � ! � & � � �
�� ! $ � � % � � " � V � � � �� < � � �. � � 4 � # � � � � � $ # � � � � $ � � - � " � ! � � & '$ I � � 2 � � �.
�� � � � � � � ! � � � ! � & I � � 2 � � � V � � � �DISCRETE (* = � � � � CO N TIN O U N S
� � � � � � � � � : $ � � = � � � � @ � � - � � $ � - , E ? J , � � & I � � 2 � � � V � � � � � .
� ) �1 � 2 � 3 � ... V � � � � < # � $ # ) � � > � � � � � � :
P � � D � � � : % � ! � $ I �= � 9 $ % � & ' � � @ � � ' � � P � �� � � Q � � � � * E & C � � 1 � $ # � � � � � � . @ � � P " � � � � � " �! 2 Q � M � 2 � � & ' : � �� � � � " � � � � � � ... c � �
�� � �� � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
58
� � � � � � � � � : � ) � � � � Q � � E � $ � - J � , E ? J , � � & I � � 2 � � � V � � � � � .
� � 0 �� �$ � � � � � � � � � � � � � � � � � � � �. � = � � � � V � � � � < # � $ # ) � � :
P � �� � @ � " �� � N � Q � M 4 # � � � � � $ � 2 � � " �� � � � �. P $ � � � � � $ � # � � � � � � . P 0 � � � � Q � 9 & ' & I � � � � � � � @ � � = � � 4 * � � � � � �. P � E @ � $ % � � � � * E & � � ! � � � 1 % # � � � � � � � � � � � � �� ) � ...c � �.
� � � � � � �� ��� � � � ��� �� � � � � �� � �� ��� eD i s c r etP r o b a b i l i t y D i s t r i b u t i o n s
� ! " � �� � � � � $ % ! � � � � H I �� % � � � ) � % � � �% % ! � : % ; ' % D . � �% � � , 9& � �� � � � � � �� �� % � :
6 5 4 3 2 1 � � � �
1/ 6 6/1 6/1 6/1 6/1 6/1 � � � � �
' . � � � & � � B � � � � � � � � � �% ! � % & � . � �% � � , 9 ��! � : % �% %& � �� � � � � � �� �� % � � ! � �� � � � � $ % ! � � � � H I �� % � � � ) � % � � :
�������� �� ��� � � ��� � � � 2 3 4 5 6 7 8 9 1 0 1 1 1 2
� � �� � �
361
362
363
364
365
366
365
364
363
362
361
59
" � �� � � � � � & I � � � 2 � � V � � � � � - � � C � J , � � � � � � � , . � > � � � � � � & I � � 2 � � � V � � # � & � �� � � � � > � � � � �� < � � �� � � / �% � � � � ; ' � , ! . � :
� � � � � � � � � � � � � .)F.D.P(Pr o b a b i l y De n s i t y Fu n c t i o n > � � � � V � � �% � � �! � , 9X $ 1 # � E � �� - , E ? xn ... x3 , x2 , x1 , � �� � � �� P (X=xi) 3 � i = 1, 2 ...n : � � � �� < � � � �� � � � � � , . � ; '
�� � � � � � �� � � � � $ ' �) ! � � $ � �� � � � �� � � �� f(xi).
> � � � � � � V � � # � � �� � � � � $ ' �) ! : � � , E ? � � X& � �� � � � ! 2 � � : x2 … … … … ..xn x1 X = x
f(x2) … … … ..f(xn) f(x1) f (x) = P (X= x)
� � � � � � ! � � � � � " # � � � � # � � � � � � � � � � � $ � % � & � � % ' � � � ( � ' ) � � � � � � # � � * � � + � � � � % " # � � � �� � �
�� , � � � - � . � � � � � � $ � % � � � � � �� , � � - � � � � � # .
� � �� � � � � � � � � � � � $ � � 0 � 1 - � �+ ( 2 0 �� - � �x � � � ! � ( ! � � � � � $ �x � * � � - � � � � � � � ( � � � � � � *
P(X=xi ) & � � � � , & � � � f(x) � � 3 # � � ( * � � � � � � :
1)x(f ii)
0 (x) f )i
in
1i=
≥
∑=
6 0
� � � � � �1 : > � � � � � � & I � � 2 � � � V � � � � �% � � �! � , 9X � � = � � � " � � � � . �
� $ � % $ � � - & � % � $ � �� ! � � : � < # � � ; ' � � : & % � � � L � 1 � � � � $ % ! � � � � " � �� � � =}5 5 � K = 5 � 5 K = � K = K ={
� $ % ! � � � � � � � �X & . 0 � 1 � 2 � , �
1/4 2) X(P (2) f1/2 1) (X P (1) f1/4 0) (X P (0) f
x) (X P )x(f
===
===
===
==
� � � & � �� � � � � > � � � � � � � ! �X& # �� ! � . $ % ! � � � � : 2 1 0 X = x 4/1 2/1 4/1 f (x) = P (X= x)
& # �� ! � % � � � � � � � , . � � � ! � � :
1
4/3
2/1
4/1
0 X
1 2
�������� �� �� � �
�����
����
6 1
� �������� �� ��� ������ D i s t r i b u t i o nP r o b a b i l i t y � � � � $ � � � & I � � 2 � � � V � � � � , E � � �� � � �� � � ' � � � % � 4 � �V � � & ' �% % ! �
� $ % ! � � � � $ � - � �� V � � � � , E � � �� � � �� (�C � � � � % �� %X� � - � $ � � � � � � & � �� � � � � > � � � � � : � � � �. � � % $ � � � � D , . � $ % ! � � � � $ � - � � �� $ � - Q � �� �
V � � # �X.
� �� � ���� ��������� � �� �� ���C o n t i n u o u s P r o b a b i l i t y D i s t r i b u t i o n s
: % ? � � = � � & I � � 2 � V � � � & � �� � � � � > � � � � � � � % � � (�C � �% % ! � % � � � J , � � $ � I �� � � � � � � � � � �� � � � � � � = � � � � V � � � � � , � � $ 1 # � E � � � � � � � �� � � � D , . � � $ � - � ! � S �E � �.
� � � � �2 : % � �� 0 ! � ! : % � � � � � � � � � �� � ! � d � � $ ! % �! � $ � W �% � � � F 1 % � � � � � � # � E $ ! % �! � � � � � � � � � � � & ' � # E � � � � ! � � � � � � � � � � � � � �
�% . 0 ! � � � � � ' � E W < � 9 0 ! � � & � �� � � � � � � � � = � � & I � � 2 � V � � � � � & � � 100 $ � � � � D , � � D + � # � � 0 ! :
� � �� � � � � � � � � � � � $ � % � & � � � �+X ( ! � � � � � $ � � � X ( � � � *
� � � 5 � *x � * x) X(P ≤ & � � � � , & � � � F(x)( � - � �
)xi(f x) P(X (x) Fx xi
∑≤
=≤=
6 2
� � � � � f (x)
� � � � � � � �)� � � � � (
! " � # $ � � � � � � % � � & X
0 .0 1 1 0 .9 0 0 .0 7 7 0 .9 5 0 .25 25 0 .9 9 0 .32 32 1.0 0 0 .30 30 1.0 1 0 .0 5 5 1.0 5 1 . 0 0 1 0 0
� � � $ � - � � % � � �% � � # ' �� [ � � � � � � � � I � � 2 � 0 ! � � E � � �� � � % � � � � - J � � �0. 9 5 � 1. 05 & I � � 2 � � � V � � � � � , � � $ % ! � � � � � � � � � � �' � � � � � � � ! � . � = � � � � V � � � � $ � �� & ' $ # ! 2 � � � H � �� % � � �% % ! � � 5 � , � �
> � � � � � � V � � � � : � �� & ' � �� � � � . �� ! . $ � �� � � � � � � 1 � � �' � �� � ! < # �� E �$ # = � � � � $ I � � 2 � � � " � V � � � � : � �� & ' " � �� � � � � � � � � �% % ! � � �� = �. � ! �X Q � � � & ' > � (* = � � � V � � (a.b) J � b x a ≤≤ � ! 2 � � & ' �� ! & � �� � �
f ( x )
)dxc(p ≤≤ a c d b
x
6 3
� = � � � � & I � � 2 � � � V � � � � > � � � � �� � � � � �'x Q � � � & ' (a, d ) � � � � � �� : �d) X c(P ≤≤& # �� ! & � � � :
d) x c(P ≤≤ = $ � � � � � � � � = � � � � $ � �� � � �f(x) & � ' @ � � � � � � � K � � � � � � �x = d , x = c � ! 2 � � & ' $ # # / � � � $ � �� � � � J � .
� � ! �f(x) � = � � � � V � � # � � �� � � � � $ ' �) ! $ � � X � �! � , 9 : i) f(x) ≥ 0 : � � � f(x) ii) ∫
Rf (x) d x = 1 1 = < % � % � � � " � � $ � �� � � � � � �!
∫b
a f (x) d x = 1
(a b≤×≤ )
∫∞
∞−
f (x) d x = 1 ( ∞≤×≤∞− )
� ��������� ���� � ����� ���� � ���� � � �� � ��� � �� ��� � � ? � � �� � � � � & .X J � �� � � � � � - � a J � a) x(P ≤ �� � � � � � � � ��(x) F � , ! . �
dx (x) f a) x( P )x(Fa
-∫∞
=≤=
6 4
& � �� � � � ! 2 � � & ' �� ! & . � :
������3: > � � � � � �� � � �X Q � � � & ' )a, b (� �� J
dx)x(f b) x a(Pb
a∫=≤≤
=
= f (b) - f (a)
� K � � �� � :
f(x)
∞−f(x)
x= a
f(x)
X
x
F(a)
F(b )-F(a)
∞−f(x) ∞
f(x)
a b
dx )x(f dx)x(fa
-
b∫∫∞∞−
−
6 5
���������� ���� � � ����� � �
1P � �! � , 9c � " � �) x& . : � �� � � � : � � & I � � 2 � V � � f (x) = c ( )5x x = 0 , 1 , … , 5
" � �) � � $ � - K K � ? 'c 2P � �! � , 9X " % �! � �� $ % � & ' � � � 3 � � � � � ) � & I � � 2 � V � �
C � �� � � � � : � � � " � �) X $ � �� � � � � = � � < # � & . : X = x 0 1 2 3 4 5
f ( x) = P ( X=x) C 2C 3C 4C 1.5C 0 .5C
� : � � �� � � �� � � � C �� � �� :� � :
i) P ( x < 3 ) ii) P (0 < x ≤4 ) iii) P (0 < x < 2 )
�� �� : � � ��� � � � � � � � � � � � � � � �X
6 6
3P " % �! � $ � �� ! � � : � < # � � � = � � � � � < � � $ � % $ � � - " � � � � , 9X � � < � � � % � � $ � � - �� � " � � � & � � � " � � � � � � ) � & I � � 2 � V � �
$ � �� ! � � : � < # � � � = � � � . V � � � � � �� � � � : � � � �X. 4P � � � � � : � 2 � � � � � E � �)" � �! � � ( � � $ � � � � � �10
& I � � 2 � � � V � � � � � �! � , ; ' : � 2 �X � ? ' �� E � � � � 2 � � � - � ) � K � � �� � � � � : � � X .
5P � �! � , 9X : � �� � � � $ ' �) ! & I � � 2 � V � �
f(x) = 21
�2 x 0 ≤≤ K K � � :
i) P ( 0 .5 < x < 1.5 ) ii) P ( x > 0 .25 ) iii) P ( x < 0 .75) iv ) P ( x > 3 )
6P " % �! � , 9
2x2)x(f −
=
0 < x < 2
6 7
K K � � :
i) P ( 0 .5 < x < 1 ) ii) P ( x > 1.5 ) iii) P ( x < 0 .3) iv ) P ( 0 < x < 2)
7P � 1 2 � � " �% �� � � � � � � & ' � �� � � � � �! � , 9 � $ # I � � $ � � E � � � � ! � � � $
� . $ � � = : � � � � � N � J � < # � � % � � � � � � � � 4 � �0. 6 . : � � 2 ' � � � � � - $ � � = " �� � 9 �� # � 4 & � � � $ # I � @ � � � �% � � �
� � � �� $ � � M � & 'x. > � � � K K � �x & � �� � � � � .
8P & . � �- � $ ) * ) : # � 4 � � 2 � 3 � 4 � � , 9 � � � 4 � � � � �%
�% C ' � , 9 � $ ) * ) � � � �- @ � � � < # � � � � ! & ' 2 N � � � 2 N 3 � � � �x � � �% � � � � - � � B � � � � ) � � . H � % � � � � $ % ! � � � � H I �� % � � = � �
> � � �x& � �� � � � � . 9P � . + �= � M � � �� � � � & ' 4 � �� � � U % � � � �� � � � � �! � , 94
1 � � � � � �� � � * �4 * � $ ) * ) . � ? � �% C ' � , 9x � � �% � � � � ) � � . = � �
K � & � �� � � � � > � � � � � H � % � � � � $ % ! � � � � H I �� % � �x .
6 8
�� � � � � � ������ � �
1P
1/32 C 1 ) 15101051 ( C
1 )( C 1)( C
1)x(f
5x
5x
x11a
=
=+++++
=
=∴
=
∑∑∑
2P � : ∑ =
x11a1)x(f
∴ C + 2C + 3C + 4C + 1.5C + 0 .5C = 1 ∴ C = 1/ 12
��� � �� : i) P (x< 3) = P (x=0 ) + P (x =1) + P (x = 2)
= C + 2C + 3C = 6 C = 6 (1/ 12)
= 1/ 2
ii) P (0 < X ≤ 4) = P (X=1) + P (X=2) + P (X=3) + P (X = 4)
= 2C + 3C + 4C + 1.5C
6 9
= 10 .5. C = 10 .5 (1/ 12)
= 10 .5/ 12 � �
= 1 – [ (P (X=0 ) + P (X=5)] = 1 – (C + 0 .5 C) = 10 .5 C = 10 .5 12
iii) P (0 < X <2) = P (X =1) = 2 C = 2/ 12 = 1/ 6
��� �� : � � ��� � � � � � � � � � � �x
F(x) = P (X ≤ x) F(0 ) = P (X ≤ 0 ) = 1/ 12 F(1) = P (X ≤ 1) = 3/ 12 F(2) = P (X ≤ 2) = 6 / 12 F(3) = P (X ≤ 3) = 10 / 12 F(4) = P (X ≤ 4) = 23/ 24 F(5) = P (X ≤ 5) = 1
70
3P f(x) = (1/ 2)x x= 1, 2, …
4P X = x 1 2 3 … … … … .10
f ( x) = P ( X=x) 1/ 10 1/ 10 1/ 10 … … … .. 1/ 10 f(x) = 1/ 10 x = 1, 2, … , 10
5P i) P (0 .5 < x < 1.5) = ∫ .5.1
5.0dx 2/1
= 1.5
0.5
2x
= ½
ii) P (x > 0 .25) = ∫2
25.0dx 2/1
= 7/ 8
iii) P ( x < 0 .75) = ∫75.0
5.0dx 2/1
= 3/ 8
71
iv ) P (x > 3) = ∫∞
3dx 2/1
= 0 0 ≤ x ≤ 2 � �
6P i) P (0 5 < x < 1) dx
2x2
1
5.0∫ −
=
1
5.0
21
5.0
1
5.0
1
0.5.
0
1
5.0
4x - x
dx 2x -dx x
dx 2x - 1
=
=
=
∫ ∫
∫
= 1/ 2 - 165
163
=
ii) P (x > 1.5) dx 2x2
2
5.1∫ −
=
= 161
iii) P (x < 0 .3) dx 2x2
3.0
0∫ −
=
= 0 .2775
72
iv ) P (0 < x <2) = dx 2x2
2
0∫ −
= 1
7P x 0 1 2 3 4 5
p(x) 0.4 ( 0.6 ) 1 ( 0.4)
( 0.6 ) 2 ( 0.4)
( 0.6 ) 3 ( 0.4)
( 0.6 ) 4 ( 0.4)
( 0.6 ) 5
8P x 4 5 6 7 8
p(x) 91
92
93
92
91
9P P (x) = n
xC p x q n-x
P (0 ) = 6427 , P (1) =
6427 , P (2) =
649
P (3) = 641
73
�������� ��������
74
������ ���� ���� ��� ������ �� ���� ���
���������� � ������ �� � ������� �� "��#���� F�� �� �� �!�)"�� ������ ( "�1 = � % ��
� . � ��� � F��� �� C �� : I �= � � � "���� ��� ��� % �� � &I �= � M � > �� ��� 0 �� � � . � ������� � > �� �#� > C � ��� � !� � � . � � ��� : "��#���� D , .
. �V � � �� ���� * � 0 �� � � . � � ��� ��� � �2 � % � � "��#���� D , . $ � � � � �% � % &# ��' � :
� ���� �� Expectation ��! �, 9x : ���� � � $ ' �) ! &I �� 2 � V � � f(x) � � ��� � �'
∫∞∞-
dx f(x) x > - � � ��� . (�� ��) �� !� � � ��� �, . �) � � � � ��!� : % � ( < #� �� � : % !�� "��� ) �� � ��� � � � $ ' �) ! $ �� �� % �C � � &� ��f(x)
V � ��� ��! �, 9 �x J � �� $ ��� �� D , . &' > - � � �� U �= ���� � � ∑ xall
P(x) x
3 �P(x) V � �#� $ ���� � � � $ �� �� x > - � � � � �% �x � ���� E(X) � � �� 3 � E $ C � $ #�� < �9 2 Q � $ % �� V � ��� < #� x �� % � �C �� :
75
E (x) ∫∞
∞
µ==-
dx (x) f x
� � µ== ∑ xP(x)
xall
V � ��� �� ! 4 � �x(���� � � � � * = � � . > - � � �� < � ��!E(x) � � �� � E � � � &���� � � � > � � � #� &��� � �� �� � ���
&� � M �µ) � � ( �� ��� #� 4 ��V �� &' V � �#� $ �- � � ��� $ �� ��x. > - � � �� $ V = �� / � * �)(* ) � > �� � ��� V � ��� $ ��� &' ( : �2 � �� % �
(�% �� � �P(x) �1 % �� % � $ � #�� xi) 2 , 1 � 0 � =i ( 4 �� � � ��� > - � � ��� &. � ��� � @ � D , . � � ) � !���� � � � : �� %.
��! �, 9x &' $ �� J � ��' (�I �� 2 � (�V � � x) (* ) � � � % � (x) ϕ ( &.J � �� : �- � � � &I �� 2 � V � � Q E @ � :
E [ ϕ (x)] = P(x) (x) xall∑ ϕ
= ∫ϕ xRange
dx f(x) (x)
�� ! 4 � �x (* = � � � � (���� � � .
, K K E ? % � r x (x) =ϕ �� K K !
E (xr) = ∑ xall
r (x) P x
76
= dx (x) f xRx
r∫
� � % �� r = 1 �� K K !
E (X) = ∑ µ= xall
(x) P x
= µ=∫ dx (x) f xRx
: 1 � � � � � � � �� � . ��' $ ��� � � "�� 3 * ) $ % �� � % $ ��- "� �� &C ��� > - �
� � / "�� ��"� � ="Y
������ &��� �� � � �� &' � / �% ��� "� ��� � � �� &% ��� L �1 �� � ) �% �� �% % !�:
� � �� � � ��� � � � � � � � � � � � � � � � S S S 3 8/1 5 S S 2 8/1 S 5 S 2 8/1 5 5 S 1 8/1 5 5 5 0 8/1 S 5 5 1 8/1 5 S 5 1 8/1 S S 5 2 8/1
77
B X "� �� �� % � D � = : � < #� � � = � �� "�� � � ) �� V � �#� � ��� � � � : �� "�� 3 * ) � % �� : ��- X &��� �� � � �� &' D ���� :
X = x 0 1 2 3 f ( x) = P ( X = x) 8
1 83
83
81
E(X) = ∑
=
3
0 i(xi) P Xi
= 0 (81 ) + 1 (
83 ) + 2(
83 ) + 3(
81 )
= 23
&% ���� � � ��� ���� �� � � �� &' $ % ���� $ ���� � � � $ �� �� � ) �% �� �% % !����� ��& :
f ( x ) = P ( X = x )
• •
•
8/3
8/2
0 1 2 3 X
78
� �������� �� P r o p e r t i e s o f E xp e c t e d V al u e
> - � � �� � � �� �% � E : % �� $ C � $ #�� < �9 2 � � �� � ��� S �� E ��� � � �� �, . � �� �� � � �� �, � � &��� �� &I �= � M � V � ��� < #� J �
$ ��� �� :
� � � � �a � � � � x � � � � � � � ! � � E (ax) = aE (X)
��K K K . ��� :
E (ax) = P(x) ax xall∑
= ∑ xall
P(x) Xa
= a E (X) � �
E (ax) = dx (x) xfa-
∫∞
∞
= a dx (x) f -
∫∞∞
= a E (x)
������ ��� ������ � � ��! �, 9 a � "��) X �; ' &I �� 2 � V � � E (a) = a
79
��K K K . ��� :
E (a) = P(x)a xall∑
= ∑ (x) P a = a (1) = a
$ K K K � %:
E (ax+ b ) = a E (x) + b ��K K K . ��� :
E(ax+ b ) = E (ax) + E(b ) = a E (x) + b
� �� � ��2 : E(2x + 3) = 2 E(x) + 3
" � � # " � # " � � $ � % #
E [ E(x)] = E (x) ��K K K . ��� :
"��) �� J � �� "��) �� > - � � �� ���� �% !, ��! $ � ��) $ �- > - � � �� �� ���> - � � �� J � �� > - � � �� > - � � �, �.
> - � � < �9 �% � � � #' E(x) $ � ��) �� $ �� ��� z ��' E(z) = z �� % �� E[ E(x)] = E (x)
8 0
4& E [ ( X - E ( x) ] = 0 ��K K K . ��� :
E [ (x – E (x) = E (x) - E(E (x)] = E (x) - E (x) = 0
�������1 : � � �X � Y � ��� � � � � ���� � � : E ( X ± Y) = E ( x ) ± E ( Y)
� � � � � � � � � � � � � ���� � � � � � � � � � � � � � � ���� � � � � ��K K . ��� :
E (X ±Y) = ∫ ∫∞
∞−
± xdyd f(x.y) y)(x
= ∫ ∫∫ ∫ ± dxdy y)(x, f y dxdy y) (x, f x = ∫ ∫∫ ∫ ± dx]dy y)(x, f[ y dx]y)dy f(x, [ x = ∫ ∫± dy f(y)y dx f(x) x
= E(X) ± E(y )
�������2: � � � Y , X � � � � ���� � � � �! " � � � ��� : E ( XY) = E ( X) E ( Y)
��K K K . ��� : E(X Y) = ∫ ∫ y d x d y) (x, f Y X
8 1
= ∫ ∫ y d x d f(y) (x) fxy … … . � * � � � �
= ∫ ∫ y d (y) fy x d (x) fx
= E (X) E (Y)
� � � � � �3 : ��! �, 9X&. : ���� � � � : � �� �I �� 2 � �V � � :
f(x) = 2x 0 ≤ X ≤ 1 �� �
�K � � K ! K K � ? ' E(x+ 1)2 E(x2) E (x)
� � � � � : E(x) = ∫1
0dx (x) f x
= ∫1
0dx (2x) x
= ∫10
2 dx x2
= 2 1
0
3
3x
= 2 0-31
= 32
8 2
E(x2) = ∫1
0
2 dx (x) f x
= ∫1
0
3 dx (2x) x
= ∫1
0
3 dx x2
= 2 1
0
4
4x
= 21
E (x + 1)2 = E [ x2 + 2 × + 1] = E(x2) + 2 E (x) + 1 = ½ + 2 (2/ 3) + 1 = ½ + 4 / 3 + 1 =
617
� �������� ���� �� ������� Variance & Standard Deviation
"� K 2 � #� 0 �K � � &K . J �K ���� � �K � % � �� ��K �� �� �� � �%(D is p e r s io n ) "� K 2 � #� 0 �� �! �O � ��� 2 - �% % � �% % !�� J �!� > � � � � � �
&I �� 2 � V � �� &���� � � � > � � � #�. : �- � � ��� $ �� �� ��) &��K � � �� �� � �� � � (� � � K V � �� &��K �� � � > K �
�K � $ �� "��� #�� �% ��� � &���� � � � > � � � �� � !� < #� �% � � �� &I �� 2 �
8 3
� �� % !�� � ) !� �� $ � � + � 9 &' �� � � �� �, 9 ���� Q ��� &' �� � �� : K � � �� &I �� 2 ��� V � ��� � - "� 2 � � � �2 � % � Q � �� "��� #�� J � �% ���
Q E � < �9 . ) !� < �9 �% . ��� % � � 0 �K � � �K � � ���� K � �� (��� 2 0 �� ��� J ����� � �� % � �� ���� �� &. � &I �� 2 ��� V � �#� &���� � � � > � � � �� "� 2 �.
$ K � � 0 1 % K � ����� 0 �� &I �� 2 � V � �� &���� � � � > � � � �� ���� �� > �� �� � � � % �O � �% % � � . � � �� �1 ��� J �!� �� > � � � �� $ ��� &' : � �-
"�' �� % � �K � "�' �� % � � > �� �� � � � � 9 �� � � $ �- � � ��� $ �� �� �� � � ��J �!� �� > � � � �� � � $ % �#� &��� � �� �� � ��.
- �� � ������ � � ��� � ��� ��� � ����� � � �� ���� �����X � ��� � � ��� �� � �� � � �����X ��! � " �� ����� " ��# �� �� ��� ��! � � ����� � � �� X ��$ �% & µ E ( x ) = ����� ' �� ��� �� �% ( �X � � σ2
x �) V a r ( x ) �� σ
2x = V a r ( x ) = (x) p )x(
xall
2∑ µ−
= ∫∞∞
µ−-
2 dx (x) f )x(
� �) ��� # �� ��� ���� ��$ * � � +, - �
= E [ ( x -u ) 2] = E [ x - E ( x ) ]2
8 4
�� �- � � �� � � �� "�' �� % � > �� > - � � �� � ��� � . ���� �� �� J �.
� - �� / � * % X K � $ ����� � � �� &. ���� �� 4 �� � &' $ #E � �� X �, !. � � ��' � - 0 1 % � "� 2 � �� 0 � � ���� � X < #� � = � % &!� $ ����� �� �� ��% 9 �
� - 0 1 % � "� 2 � #� 0 �� �X �� � . � ���� #� 4 � ��� &��� �� , �� % �% % ? ' J ����� � �� % � �� < � .
J �K ���� � �� % � �� ���� �� 4 �� � �� < �9 � � ���� 4 * ��� U = % %C ��� � �- J �!� �� > � � � #�+ � �� �, . &' �� - &.
������4 : ��)� <�9 � �����)1 ( B � " � K � 3 * ) � % : �� - + �� �9 �. � x & K .
� �= : ��� �� / " � � � & I � �2 ��� V � ��� . $ K �� & K � � & ��� �� �� �� V � �# � ���� � � � x � ��� �� � � �! 4 �� � $ 1 ! � 2
xσ J �K ���� � � � % � � � xσ .
- � � � � � � � � �
������ ������ �������� ���� � X ���� ���� �% . �� �� � ��� �� * . ����X � ����� / � � ���� xσ ���� �� :
xσ = 22x )-(x E µ=σ
85
X = x f ( x) = P ( X=x) Xf ( x) ( x-E ( x) ) 2 [ x-E ( x) ] 2 f ( x)
0 1/ 8 0 ( 0-3/ 2) 2 = 9 / 4 9 / 32 1 3/ 8 3/ 8 ( 1-3/ 2) 2 = 1/ 4 3/ 32 2 3/ 8 6 / 8 ( 2-3/ 2) 2 = 1/ 4 3/ 32 3 1/ 8 3/ 8 ( 3-3/ 2) 2 = 9 / 4 9 / 32 E ( x )
23
812
=
43
32242
x ==σ
� � ? ' � , ! .43 2x =σ� ��� ��
43
x =σJ � ���� � � � % � �
23 = ������� : � � �X � �� � � � �� � � �� � ����� � � µ = E (x) � ��� � � �
2Xσ = V a r (x) � �
2σ = E (x2) - [ E (x)] 2 = E (x2) - µ 2
� �K K K . ���: 2σ = E [(x - E (x)]2 = E [(x2 – 2xE(x) + { } ])x(E 2 = E (x2) – 2{ } { }22 )x(E)x(E +
= E (x2) - [ ]2)x(E
86
�������5 : � % : �� - & � �. � � ��� �� ��)��� <�9 � �����3 �� K �� � �K ' " � �
K � J � ���� � � � % � � � � ��� �� 4 �� � : � � & � � & ��� ��x $ / % �� � � E � � �� $ � ��� ��
[ 2σ = E (x2) - µ 2
X = x f(x) xf(x) x2f (x) 0 1/ 8 0 (0)2 = (1/ 8 ) = 0 1 3/ 8 3/ 8 (1)2 (3/ 8 ) = 3/ 8 2 3/ 8 6 / 8 (2)2 (3/ 8 ) = 12/ 8 3 1/ 8 1/ 8 (3)2 (1/ 8 )= 9 / 8 E(x)
23
= E(x2) = 824 = 3
J ��� � ��� �� � �' � , ! . �
2σ = E(x2) - [E (x)]2
= 3 - 2)23(
= 23
� ��� # � 4 ���� & � � � �� , �� J ��� J � ���� � � � % � � � =23
43==σ
87
� �������� ����� Properties of Variance
� � �% # � �� � � S � �E �� D , . � � ��� # � S � �E D � 5 �% . J @ � ��� �� 4 � & I � �2 ��� V � ��� & ' : �� X& . S � �E �� � . � � :
1( � � � a � � � �� X � � � � � � � � � � � � � V ar (ax) = a2 V ar (x)
� �K K K . ��� : V ar (ax) = E [ax – E (ax)]2 = E [ax – a E (x)]2
= a2E [x –E (x)]2 = a2 v ar (x)
������6 : � ��� � �! � , 9x J ��� 0 . 5$ ��� �� " � V � ��� � ��� �. ��' :
i( 2 X ii( 2
X
���� � � : i) V ar (2x) = 4 V ar (x)
= 4 (0. 5 ) = 2
88
ii) V ar )2
X( = 41 V ar (x)
= 41 (0. 5 )
= 0. 125
2 ( � � � � � � � � � ��� � � �� � � �! � , 9a � �' " ��)
V ar (a) = 0 � �K K K . ���:
V ar (a) = E [a –E (a)]2 = E (a – a)2 = E (02) = 0
$ K K K � % : V ar (x ± a) = V ar (x)
� �K K K . ��� : V ar (x ±a) = V ar (x) ± V ar (a) = V ar (x) ± 0 = V ar (x)
K V � ��� � � �� <�9 : � ��) $ � - J � [ � �� $ � - J � $ ' �C 9 � �' � , ! . � V � ��� � , � � � ��� �� <# � )N � � & I � �2 ���.
89
������7: V � ��� � ��� � �! � , 9 x J ��� 5 � � �! � ��� �. ��' :
i) x + 3 ii) x – 6
���� � � : i) V ar (x + 3) = V ar (x) = 5 ii) V ar (x – 6 ) = V ar (x) = 5
3 ( � � � X ! Y �� � " # � � � � � � � � � � � � � � � : V ar (x+Y) = V ar (X) + V ar (Y) V ar (x - y) = V ar (x) + V ar (y)
� J ��� ��� % � � 1 �� �� � # � � � � � I � �2 � � V � � B �� � � ��� � � J � B �� ��� % ���.
� �K K K . ��� : V ar (X ± Y) = E [(X ± Y) – E (X ± Y)]2
= E [{ } { } 2]E(y)- Y )x(EX ±−
= E [{ } { } { }{ })y(EY)x(EX2E(y)-Y )x(EX 22 −−±+− ]
= E { } { } { }{ }])y(EY)x(EX[ E2E(y) - YE )x(EX 22 −−±+−
= V ar { }{ }])Y(EY)x(EX[ E2(Y)Var )x( −−±+
9 0
� # � � � � � ) �� �� � � ��� 1 = J ��� E @ � � �� � ? � " �)% � � �% # �' � , ! . �
(� ��� � E �� � �� � �! = 2 [ E { } { }]E(y)-YE )x(Ex − = 2 [(0) (0)] = 0
∴ V ar (X± Y) = V ar (x) + V ar (Y)
J � : # � � � ��� " � V � ��� � � � � � � � J � <# � D � �� �� D , . � ��� � ! � � � � :
V ar (x1± x2 ± x3 ± … ± xn)=V ar (x1)+V ar (x2)+… + V ar (xn) � ����������� variance C o
� V � � � � � ��� * � 0 � � � �. � Y, X " % �! � , �' f(xy) ���� � � $ ' �)! : �� � V � ��� �V � � �' ��� � V � ��� � , . Y, X & # ��! (� C � � � :
Co v (X, Y) = ∫ ∫∞
∞−
[X –E (x)] [Y – E(y)] f (xy) d xd y
= E [{ }{ }E(y)-Y )x(EX − ]
9 1
� ���������� � ������ � 1� � � � � � X, X � � � � � X � Co v (x, x) = V ar (x)
� �K K K . ��� : > C ��X � � Y 4 �# � ��� <�9 �= % D * � � �V � �� : � ��� & '
2� � � � � � � b , a � � � � � � � � : Co v (ax, b y) = ab Co v (x, y)
� �K K K . ��� : Co v (ax, b y) = E { }{ }[ ])by(Eby)ax(Eax −− = ab E { }{ })y(EY )x(Ex −− = ab c o v (x, y)
3� � � � � � � a � � � � � � Co v (x, a) = 0
P ���������� ������ � � ������� �� �� � � �Y, X) � � ���� � � � � �� � � ���C
� �Co v ( ������ �� � � ��� � � � �� ! �� " # � � $ % ��� $ � � � � �& � � � �� ' �� � �� $:
Co v ( x , Y) = E [ ( x -µ x ) ( Y-µ y ) ]
9 2
� �K K K . ��� : Co v (x, a) = E { }{ }[ ])a(Ea )x(Ex −− = E { }{ }[ ]0 )x(Ex − = 0
4P Co v (x1 + x2 , y) = Co v (x1 , y) + Co v (x2 , y)
� �K K K . ��� : Co v (x1 + x2 , y) = E { }{ }[ ](y) E-Y )xx( E)xx( 2121 +−+ =E { }{ }[ ] { }{ }[ ])y(Ey )x(Ex E)y(Ey )x(Ex 2211 −−+−− = Co v (x1 , y) + Co v (x2 , y)
5� � � � � � � � � � � � � � � x2 , x1 � � � V ar (x1 + x2) = V ar (x1) + V ar (x2) + 2Co v (x1 , x2) V ar (x1 - x2) = V ar (x1) + V ar (x2) - 2Co v (x1 , x2)
� �K K K . ��� : V ar (x1 + x2) = E [ (x1 +x2)
- E (x1 +x2)]2 = E { } { }[ ]22211 )x(Ex )x(Ex −+− = E 2
222
11 )]x(Ex[E)]x(Ex[ −+− + 2 E { } { }[ ])x(Ex )x(Ex 2211 −−
= V ar (x1) + V ar (x2) + 2 Co v (x1 , x2)
�� & ' J � � V � � � � � 1 �� � ��� : ��� & ' �)�����: �
93
V ar (x1 – x2) 6� � � � � � x2 , x1� � � � � � � � � � � � � � � � �
Co v (x1 , x2) = 0 � �K K K . ��� :
Co v (x1 , x2) = E { } { }[ ])x(Ex )x(Ex 2211 −− = E[x1 x2 + E (x1) E(x2) – x1 E (x2) – x2 E(x1) = E(x1) E(x2)+E(x1) E(x2)–E(x1) E(x2)-E(x1) E(x2) = 0
� �! � # � � � � � V � ��� � � ���� E (x1 x2) = E (x1) E (x2)
� ����� � � � � �� � � � �
� � �V � �� � �! � , 9 � 1 = ������� � , . � �! �Y , X 1 = J ��� . Q ��� & ' > � � ��� � � ������
1 ≥ ρ ≥ - 1 � ; K ' � , K ! . �
ρ 2 ≤ 1
� � � � � � � � � � � � ρ= (y)Var (x)Var
Y) , (X Cov
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Y , X
94
� �� � � � � � �� � � � � � � � �
1 ( � �! � , 9µ E (x) = � � " �)? ' : i) E (x-µ ) = 0 ii) E (x-c )2 = E (x - µ )2 + (µ -c )2
3 � C " ��) J � .
2 ( �� � �K = E 2 �� � � � S E 2 � � � " � ��� � � " % �! � , 93 �� 4 �� 5 �� 6 4 � � �� <# � & . � I �� # � <��@ � $ �� & ' S �E 2 � :
0. 05 � 0. 43 � 0. 27 � 0. 12 � 0. 09 � 0. 04 I �� # � <��@ � : �� & ' � . � � > - �� ��� S �E 2 @ � � �. ��'Y �
3 ( � � ��� � � " �)�E (x-a)2 " ��)�� � �! �� % � Q V = �� : � �� % & ' � �! a J ��� E(x).
4 ( � # � � � ��� � V � ��� �% � � �! � , 9Y , X& . ��� ���� � � $ ' �)! : f (x) = 12 x 2 (1 – x) 0 ≤ X ≤ 1 f (y) = 2 Y 0 ≤ Y ≤ 1
� � ��� > - �� �� Yx
xY2 +
95
5 ( � �! � , 9X & ��� � �� : � � � & I � �2 � V � � 10 : % ��� � 6 $ � - �� E (x2 + 3 x)
6 ( � � ��- � � $ � ����� " � � � �� �� & ���� � � � > � �� �� & � � & ��� �� �� ���� & �I �� � 2 �* E " � � �:
� � � � � � � � � � � � � � � � ! ! " 0. 01 0 0. 05 10 0. 39 20 0. 45 30 0. 10 40 1.00 #������$���
KK �� : i( � 2 � � � * E : � � � � � � " � � � � � � � � �� �.
ii( � 2 � � � * E : � � � � � � " � � � � � � � � � �. 7P� & ' � � � : � � � � � � : % �# � � � " � % � � 1 # � � � � � � & � � � � � � � � � " * � � � � �
� � % � � ! � / � % � � � � � � � � � � � (� � �� � � : � � ���� � � � � % � & ' � � �� ! ! �"
0. 13 0 0.27 4 0. 39 6 0. 21 8 0. 07 40 1.00 #������$���
96
KK �� : i( $ � �� � @ � " � � � � � � � � �� �
ii( $ � �� � @ � " � � � � # � & � � � � � � � � �� � � � � �
8 ( � � > � � � � 4 = % � $ - � � � S E 2 J � 2 � & K. � < K� �@ � � � I � � � U �
4000 & K. � $ K % � ) � � � � I � � � �� � % 3000 � � K� � � � � � K% 0. 005 � 0. 008Y $ - � � � � � D , � � 4 � � % � � � � � � � �. � � ' 4 � � � � < # �
9 ( � � K� � � � � E K� � � K! � , � K' D � � + � 2 � S E 2 4 . ,A �� B �� C
� � � � � � �0. 4 � 0. 3 � 0. 3 � � K� � � � K� ) � � ! � 4 � � � � < # � A 2000 � � � � � � � % B 25 00 � � � � � � � % C 3000 a K# � � � � �. � � ' � %
Y S E 2 � � � , . : � ' � � > - �� � � �
10 ( & . $ # � � � � > � 2 � $ ) * ) & ' � � � � $ ! 2C , B, A : K� - �� � � � " � I � � � � � & K. $ ) * ) � � > � 2 � # �100. 000 � 5 0. 000 � 25 . 000 4 K � � � � < K# �
& . " � % � � � � � �10. 000 � 5 000 � 3.000�. � � ' 4 � � � � < # � � % : i( $ ) * ) � � > � 2 � � � � � $ � - �� � � � " � I � � � � & � � � �.
ii( � � � � � � & � � � �.
11 ( " % � ! � , 9X � �� $ � 1 = & ' $ � � � � � � + � � E @ � � & . f (0) = 0.9 f (1) = 0.05 f (2) = 0.03 f (3) = 0.02
97
& ' > - �� � � � + � � E @ � � �. � � '200Y $ � 1 = 12 ( � � ! � , 9C � " � � ) X : � � � � � � : � � & I � �2 � V � �
f (x) = c x x = 3, 4, 5, 6
KK �� : i( " � � ) � � $ � -C
ii( > - ��X iii( � � � �X
13 ( � � " � ) �
Co v (x, y) = E (x y) - µ x µ y
14 ( V � � # � $ ! � 2 � � � � � � � � � � $ ' � ) ! : � � � � " � # � � , 9 � y, x & .
f (x, y) = x + y 0 ≤ X ≤ 1 0 ≤ X ≤ 1 KK �� i ( � V �y, x J � Co v (x, y) ii( � � � � � � � � � � �y , x J � (x, y)
98
�� � � � � � � � ��� � � � ����� 1 (
i) E (x-µ ) = E[ (x-E (x)] = E (x) – E (x) = 0
ii) E(x-C)2 = E (x2-2c x + C2) P � @ � � � � �
= E (x2) – 2CE(x) + C2 = E (x2) – 2Cµ + C2
P� � � � � � @ � E (x-µ )2 + (µ -c )2 = E (x2 - 2µ x + µ 2) + µ 2- 2µC + C2 = E (x)2 – 2 µ 2 + µ 2 + µ 2 - 2µC + C2 = E (x2) – 2 µ c + C2
∴ 4 �# � � � � �. � � �� � � � � � ' � � � � �! 2 (
x f ( x ) f ( x ) X = x 0.05 0.05 1 0.86 0.43 2 0.81 0.27 3 0.48 0.12 4 0.45 0.09 5 0.25 0.04 6
E(X) = ∑allx
)x(XP
99
= 2.89 4 �# � � � � �. �
3( J �� � � : � < � �@ � : � � 2 � � � " % � ! � , 9 Q V = � � : � � � % & ' � � � � � � �!
1 = � � 444 3444 21
Z
222 a (x) E a 2)x(E)ax(E +−=−
� � � � � � , . � � 2 %) Z ( K� $ � � % � � � a
dadz = 0-2 E (x) + 2a
= 0 – 2 a + 2a = 0
3 �a = E (x) ∴ Q V = � � : � � � % & ' � � � � �
4 ( E
+
=
+
YXE
XY E
YX
XY
22
= E (Y) E
+
Y1 E)X(EX
12
� * � � � � � � � % @ D * � � : � � � � � � & ' H � � % � � F �� % � D � < # � > - �� � ! % � O �
10 0
E (Y) = ∫1
0Y f (Y) d y
= ∫1
0Y (2Y) d y
= 2 ∫1
0Y2 d Y
= 2 1
0
3
3Y
= 32
E
2X1 = ∫
1
0{ } X)-(1 X 12 X
1 22 d X
= ∫1
0{ } )X 12- X 12 X
1 322 d x
= 12 ∫1
0(1-X) d X
= 12 1
0
2
2X - X
= 12
21
= 6
10 1
E (X) = ∫1
0{ } X)-(1 X 12 X 2 d X
= 12 ∫1
0X3 – X4 d x
= 12 1
0
54
5X -
4X
= 12 53
2012
51 - 4
1 ==
E
Y1 = ∫
1
0 Y1 (2Y) d y
= ∫1
02 d y
= 2 E
+
YX
XY2 = E (Y) E
2X1 + E(X) E
Y1
=
32 (6) +
53 (2)
= 5.2
10 2
5 ( E (X2 + 3X) = E (X2) + 3 E (X) … … … 1 V ar (x) = E (X2) - { }2)X(E … … … 2 6 = E (X2) – (10)2 E(X2) = 106 … … … 3
� � � � � 3 � � 1 E (X2 + 3 X) = 106 + 3(10) = 106 + 30 = 136
6(
�������� X – x f ( x ) = P ( X= x ) X f ( x ) X2 f ( x )
0 0.01 0 0 10 0.05 0.50 5 20 0.039 7 .80 156 30 0.45 13.50 405 40 0.10 4.00 160
E(x) = ∑
allx)x(Xp E(x2) = ∑
allx
2 )x(Px
= 25.8 = 7 26
10 3
V (x) = E(x2) - { }2)x(E = 7 26 – 665.64 = 60.36
7(
X P ( X) XP ( X) X2 P ( x ) 2 0.13 0.26 0.52 4 0.27 1.08 4.32 6 0.32 1.9 2 11.52 8 0.21 1.68 13.44
10 0.07 0.7 0 7 .00
E(x) = ∑ )x(XP E(x2) = ∑ )x(px2
= 5.64 = 36.80
V (x) = E(x2) - { }2)x(E = 36.80 – 31.809 6 = 4.9 9 04 ≅ 5
10 4
8( � � � � � � � � � � � � � � � � � � � � � � � E(x) = ∑
allx(X) P X
= 4000 (0.005) + 3000 (0.008) = 20 + 24 = 44
9 ( � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � �
E(x) = ∑allx
)x(XP
= 2000 (0.4) + 2500(0.3) + 3000(0.3) = 800 + 7 50 + 9 00 = 2450
10 ( � � � � � � � � � " � � # � � � � � � $ % � � & � � ' � � � � � � ( � � � � � � !
i) E ( X1 + X2 + X3) = E (X1) + E(X2) + (X3) = 100000 + 50000 + 25000 = 17 5000
ii) V (X1 + X2 + X3) = V (X1) + V (X2) + V (X3) = 10000 + 5000 + 3000
= 18000
10 5
11( ) � * � � � � * + , � � � � � � � � � - � . / � � � � 0 = E (x) = ∑
allx(X) P X
= 0 (0.9 ) + 1 (0.05)+2 (0.03) + 3 (0.02) = 0 + 0.05 + 0.06 + 0.06 = 0.17
� � � � � � � � - � . / � � � � 02 0 0 � � " � � � � � 1 � � $ � � � � * + , 2 0 0 � * + , � ' � � � 2 � � � * � 3 � �
= 200 (0.17 ) = 34
12 ( i) 1 (x) f
allx=∑
1 X C =∴∑
∑ = 1 X C C (3 + 4 + 5 + 6) = 1 ∴ C =
181
ii) E (x) = ∑
allxP(X) X
= ∑ CX X = ∑ 2xC =
181 ( 9 + 16 + 25 + 36)
10 6
= 943
iii) V (X) = E (X2) - { }2)x( E = E(X2) = ∑
allx
2 (X) P X
= C∑ 3X
= 181 (27 + 64 + 125 + 216)
= 9216
V ar (x) = 9216 -
2
943
= 8118491944 −
= 8195
13 ( � � � � � �
Co v (x, Y) = E (XY) - yx µµ � � � � � :
Co v (x, y) = E [ (x - xµ ) (y - yµ )] = E (xy - x yµ - xµ y + xµ yµ ) = E (xy) - xµ yµ - xµ yµ + xµ yµ = E (XY) - xµ yµ
� � � � � � � �
10 7
14 ( Co v (x, y) = E(xy) – E(x) E(y)
∴ � � � � � � � � � � � � � � � � � � � � � � � �� � � �X � � � � � Y � � � � � XY. � � ! " # � � � � �E (y) , E(x) � ! $ � � � E(x+y)
E (x + y) = ∫ ∫1
0(x+y) f (x, y) d x d y
= ∫ ∫1
0x f (x, y) d x d y + ∫ ∫
1
0x f (xy) d x d y
= 444 3444 21
)x(E
1
0
1
0dx dy] )xy(fx[∫ ∫ +
444 3444 21
)y(E
1
0
1
0dy dx] )xy(f [ y∫ ∫
= ∫ ∫ +1
0
1
0 dy] )yx(x[ d y + ∫ ∫ +
1
0
1
0 dx] )yx( [ y d y
= ∫ +1
0
1
0
2 2
y xy x d y + ∫ +1
0
1
0
2 yx2
x y d y
= dx21xx
)x(f
1
0 43421
+∫ + dy21yy)x(f
1
0 43421
+∫
= dy2yydx2
xx1
0
21
0
2 ∫∫
++
+
= 1
0
231
0
23
4y
3y
4x
3x
+++
= 127
127+
108
�� �� E (x) =
127
E (Y) = 127
f (x) = x + 21
f (y) = y + 21
� � � � � E(xy) E (xy) = ∫ ∫
1
0xy f (xy) d xd y
= ∫ ∫1
0xy (x+y) d xd y
= ∫ ∫1
0x2y + xy2 d xd y
= ∫ ∫∫ ∫ +1
0
1
0
21
0
1
0
2 dy dx] xy [ dy dx] yx [
= dy 3xy dy 3
yx 1
0
1
0
31
0
1
0
3 ∫∫ +
= xdx31 ydy 3
1 1
0
1
0∫∫ +
= 2X 3
1 2Y 3
1 xdx 31 ydy 3
1 1
0
21
0
21
0
1
0+=+ ∫∫
= 31
61
61
=+
109
� � � � � �� C o v (x, y) = E (xy) – E (x) E (y) =
127 .
127
31
+
= 14449 3
1−
= 1441
ii ( � � � � � � � � � � � � � �� � � � � � � � � � �V (y) , v (x)
V (x) = E (x2) - { }2)x(E
E (x2) = dx )x(fx21
0∫
= dx )21x( x2
1
0+∫
= 1
0
34
6X
4X +
= 125
V a r (x) = 14411
127 -
125 2
=
110
� � � � � � � ��� � � �� � V (y) = 144
11
111- 144/11
144/1
14411.144
11144/1
V(y) )x(Vy)(x, Cov )y,x( =−=−==ρ
111
������� ��� �� ���� ��
112
����������� ����� � � � � � �� � � �� � � � � � � � � � � � � ��
THE DISCRETE PROBABILITY DISTRIBUTIONS
� ������������� ! � �" � � # $ � � ��� ! �" � � � % �� � � � � � � � � �� � � � � & � � �" � � ' � ( ) � � �*
�" � � # � � � � � � � �� � � $ + � �, ! �" � � # � � �� � � $ + � �, ! � � �" � � # � � � � � ��, ! �� �% � ��� � � �� �� ! � �" � � � � � � - � . * $ � � �� �� � �� � � # � � � � � ! �" � � � )
$ � � � � � & � � �� � � � � � �� �" � � � � �. % � � � � �/ � � � � � � ! �" � � � � �� � � $ + � �,� � - � 0 � � ' �� � � � .
� � � � � � � � � � � � � � @The Binomial Distribution@@
� - � 0 � & � � �� � � � � � ��, $ � � � � �� � � �� ��� � � � � � � ��, � ��� � � � � � �� � � 2 � � �� � � ' � � ( � � �( � � � � � � * � � � � � �� � � � � �� � �� � � % � � � � $ � � & �� � � 3 �
, & �� � � 3 � � � 4 �� � & �� � � 3 � 5 . * � , � # � 6 � � � � � �� � � 3 � , � # � �� � � � �+ � �� � � � � � �7 � � � ! �- � � % � � �� $ � � � �+ � � �% � 8 �� � - � � � � � � � . 9 � � . � � � � ��
� �� � � � �+ � �, & �� � � 3 ' � ( � � � � � � $ � � � � �� � 7 � � �� � � � �+ � �, � � �� � � � � � , � � : �100 � ; � � � �3 � � � �� . 9 � � � � � � . � � � �� � � � $ � < � � � � � � � � ��
� �� � � � � � � �= � � �6 � � �� � � � �+ � �, : 1. � �� � � � � � � � � � � �. 2. � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � �� � �.
113
3. � � � � � � � � � � � � � � � ! � � � " � �� � � � � � � � � � � � # � � �.
4. $ % � � � & ' % � ( � � ) � � � � � � �. ! � � � � � � * � + �� � � , � � � � - ' �� � �) ! � � � � � " � � � � . � �
/ ) � � % � � �� � � � � � � � ( � 0 � " ( � � � ! � � � � � � � ! 0 � " ! 0 � �� � � 1 � � �� � � � � � ! 0 � " * � + �� � �.
� � ��� �1: � � � ��n ! 2 � � � � � � � � ( � 0 � " ( � � � � � ! � � � � � � � � � � ! �
� � � � ( � � � % � 3 � � ( % � 4 5 � � 6P 7 �8 3 � �� � 9 � � �8 � � � � � � � � ! � ) � � � � � ; � ( �(1-P) + � � � & � + � � � q 3 � �� � �& = / � � � � � � � ! �
7 �8)� � � � � ; � ( � � � � � ! � 3 � � � 3 � �� � �& = � � � � � ! � ; � : �� � �: � � � � � � 1 = n
� �� � ( � � � � � � � � � � � ( � �� � � � ? � )@ � A = (S
7 �8 3 � �� � 9 � 6 + � � A C � � 3 � � � 3 � �� � 9 � 6 + � � @
D �X7 �8 3 � �� � �& = � � � � � � ! 0 � �� % � � � 1 � � � � � " � � 2 � ( � � � � � � / � � � � � �� � � X ! � 0 E 1
P7 �8 � � �& = � � � � � ! � q= (I-P)� � � ! � ( � � � � � �& = � �
� 1 � � � ! � � � � � � * � + �� � � � �� � �x! � � � � � � � � � 9 � :
114
� � � � �1 � � � � � � � � � � � � � �
1 =n X = x f ( x ) = P ( X = x )
0 1
f (0) = P )�( = q f (1) = P )�( = P
� � � � � � q + P = 1
�� � � � �: � � � � � � 2 = n � �� � ( � � � � � � � � � � D �� � � � � )A A@ @ E A @ E @ A E = (S
C � � F �8 ( � � " � � � F �8 9 � �G � ! � % � A A
( � � � ( � � " � � � ( � � � 9 � �G � ! � % � @ @ ( � � � ( � � " � � � F �8 9 � �G � ! � % � @ A F �8 ( � � " � � � ( � � � 9 � �G � ! � % � A @
( � � � � � � � � � � � � � � � � ; �4 � � � ! � � � � � � 7 H � � � � � � � � �� � �
qq , qp, pq, pp ( � � ) � � � � � � � � � C � �
/ � 4 � �� � �x ! � ( � � � � � � 0 E 1 E 2
115
2 � ! � � � � � � * � + �� � � � �� � �x! � � � � � �� � � � ! � � � :
������)2( �� � � � � � � � � ��
n = 2 X = x f ( x ) = P ( X = x )
0
1
2
f ( 0) = P )� �( = P )�( P )�( = q q = q 2
f ( 1) = P )� �( + P )� �( = P )�( P )�( + P )�( P )�(
= P q + q P = 2q P f ( 2) = P )� �( + P )�( P )�( = P P = P 2
� � � � � � q2 + 2 qP + P 2 = ( q+ P ) 2 = 1
�� �� � � : � � � � � � n � � � � � 3 � �� � ( � � � � � � � � � � � ( � �� � � � ? �
}� � � � � � � � � � � � � � � � � � � � � � � � � �� � � � �{S =
( � � � � � � � � � � � D �� � � � � ; � =8 � � � ! � � � � � � 7 H & � F = � � � � � � � � � � � � �� � �
qqq, qqP, qPq, qPP, Pqq, PqP, PPq, PPP / � 4 � �� � �x ! � ( � � � � � � 0 E 1 E 2 E 3
2 � ! � � � � � � * � + �� � � � �� � �x�� � � � ! � � � ! � � � � � :
116
� � � � � �3
� � � � � � � � � � � � � n = 3
X = x f ( x ) = P ( X = x ) 0 1
2
3
f ( 0) = P )� � �( = P )�( P )�( P )�( = q 3
f ( 1) = P )� � �( + P )� � �( + P )� � �(
= q 2 p + q 2 p + q 2 p = 3 q 2 p f ( 2) = P )� � �( + P )� � �( + P )� � �(
= q p 2 + q p 2 + q p 2 = 3 q p 2
f ( 3 ) = P )� � �( = P )�( P )�( P )�( = p 3
� � � � � � q3 + 3 q2 p + 3 qp 2 + P 3 = ( q+ p ) 3 = 1
�� ( � � � � � � � � � � � � � � � � ( � � ) � � ( " . " � � � �� � � � � � � = � . � : � � � � H 6n ; � ) � 1 ( � � � � � � � � � � � � � =21 = 2 � � � � H 6n� ; � )2 ( � � � � � � � � � � � � � =22 = 4 � � � � H 6n ; � ) � 3 ( � � � � � � � � � � � � � =23 = 8
� � � 2 � � � ( 2 � 0 � " ( � � � � ( � � � � � � � � � � � � � � � �� � ( � � � � � � 7 H � / � � % � � � ; � ) �2n � � 3 � �� � ( � � � � � � � � � H 6n � � � . ! 2 � � � � � 2 ) � � � H 6
� � � � � � � � � F � � +n F �8 � � � � I 2 � � , 2 � � � � � * � � � 2 ) � ( � � � �� � 2 ) ( � � ) � � � � � � % � � � � � � ( � � ) � � ( " . " � � � �� � � � � ! � � � ( � � � � �
@ � H � � � & ) � ( � � ( � � � � � � � � � 3 � � � ( � ��.
117
F �8 3 � �� � 9 � � �8 � � � � � � � � � J � � �� � � * � � � ) � � # � �x F � � � � �� � � � � � � � � ( � 0 � " ( � � � ! � � � ' � �n � � � ) ( % � 4 5 � � � 3 2 � � (
D �2 4 � � 2 � � � � �� ( � � � � �� ( � � % � ( � � � � � � F �8 � � K � � � � �� � � � � � C � � � �) F �8 3 � �� � �& = (x ( � �4 � / � � � F � (n-x) ! � @ � H � F �
( � � � � � �. C � � � � D �4 � � � � � �)F �8 ( x �� F � Px C � � � � D �4 � / � � � � � � �n-x �� F � qn-x
� � �� � � � � � � � � K ' F � � 4 � � � C � � � � D �4 � � � � �x �� F � px qn-x
; � ) � �� � , � � � �� � � 3 � ) � � � � � � C � � � � D �4 �� ( � � � �� � � � � � � � � � � � , � � � ��x � � n ; �
! x)-(n !x!nn
x
nxC =
=
7 �8 3 � �� � 9 � � �8 � � � �� � K � � � � � � � � � � � � �� � � H � � �x F �
� � � � ' � �n 9 � 6 L � � ) � � � � : ( )nx px qn-x
� � � �� E L. 2 2 " �n ; � ) � 3 7 �8 3 � �� � �& = � � � � � � � �� � 0 E 1 E 2 E 3 ! � � �� � � 9 � ! � � � � :
(33) p3 , (23) p2 q , (13) pq2 , (03) q3 p3 , 3p2q , 3pq2 , q3 � �� � ! � & � � � 8 � ! � � � � � � � � � � I � � ! � �)3(
118
� � / � � � � � � = � . � � � �� � � � � � ( � � �
( )nx px qn-x x = 0, 1, … . . , n
� � � � � � � � � � �x � � � � � � � � � �� � � � � � (q+ p)n
� � � � � (q+ p)n = qn + ( )n1 pqn-1 + ( )n2 p2qn-2 + … + ( )nx pxqn-x + … + ( )n
1n− pn-1q + pn � � � � � � ! 0 � " * � + �� � � ! � � � � � � * � + �� � � � H � ( � � ) � � 5 � � � � � �
�� � � � � � � ! 0 � " * � + �� � � 8 � � � :
� � � � � � � � � � � � � � : � �� � � � � � ! � � " � # � �� � � �� � � � � � � � � � $x � % � � �
� & ' � � � � ( � " � ��) � � ��n * � � �� � � � �� � " � �+ � , � � - � + � � � � � � � % $ � � � � �#
f ( x) = ( )nx px qn-x . � :
i( x= 0, 1, …, n ii( p" � + � , � � 0 � & ' � � � � � � �� � � $ iii( q" � + � , � � 0 � & � ! 1 � � � �� � � $ iv( 1 = p + q
119
� � �� �2 :
! ) � � � � ! � � � � � � � � � K �)� � � � � � ( � � ! � * ' � � � ( � ) � � � � � � ) � G � ( 8 � � � � � � � � $ % � ( ) � � ( � M)� � ) � � � .( � � H 6 / % � �
� �10 % � � � � � � % � ( � M � � � � � � � �� � � � ) � � � � ) � � � � : i( ) � ; � 9 � ; �� � � � ii( � � ) � � / & � � � � " � iii( � � ) � � / & � � � � � " � � � " � � � �� �: D �x ( � 1 � � 7 H � ! � � � � �� �� � � � � ) � � � � � � � " � � 2 � ( � � � � � � / � � � � � �� � x ! � 0 E 1 E 2 E ... E20 D � p; � ) � � ) � � � � � � � �� � � � � � � � � q� ) � � � � � � � �� � � � � � � � � � ; � ) �
∴ p = 0. 10 q = 1-P = 0. 9 0 n = 20
i) p )� � � ! � " � # � # $( = f (0) f (x) = ( )nx px qn-x f (0) = ( )20
0 p0 q20 = (q)20 = (0. 9 0)20 = 0. 122
12 0
ii) P )� � � � � � � � ( = f (2) f (2) = ( )20
2 p2 q18 = 19 0 (0. 01)2 (0. 9 0)18 = 19 0 (0. 01) (0. 9 )18
= 0. 28 5
iii) P )� � � � � � � � � ( = p (x > 2) = 1- p (x ≤ 2)
= 1 - { }) 2(x p 1)(x p 0) (x p =+=+= = 1 - { }285.0.9)0( .1)0( 20 .9)0( 1920 ++ = 1 - { }.2850 .2700 0.122 ++ = 0. 323
� � � � � � � ! 0 � " * � + �� � � * 4 �� ( 2 � � ( % � 4 5 � � 6 ( � � � �� � , � ) � � � " � � � 9 � 6 F � �% � �n F 2 � � % � � �n ; � ) � 1 ! 0 � �� % � � � 1 � � � ! � � � � � � * � + �� � � � � x � � � �� �
� �� � � � ! � 9 � % � F �8 3 � �� � �& = � � �)1 (� * 2 � + �� � � � H & � * 4 �� � � � �� 2 2 2 2 2 2 2 2 2 2 � ! 2 2 2 2 2 2 2 2 2 2 � K �2 2 2 2 2 2 2 2 2 2 ) � �
)4 (! � � � � :
12 1
������)4( �� � � � � � � � � �� � � ��
n = 1 X f ( x ) x f ( x ) 0 1
q p
0 p
���������� ∑ ==µ1
0p)x(xf
� % � � �n ; � ) � 2 ! 0 � �� % � � � 1 � � � ! � � � � � � * � + �� � � � ? � x � � � �� � �� � � � ! � 9 � % � 7 �8 3 � �� � �& = � � � �)2 ( * 2 4 �� � � �) � 2 ) �� � ��
! � ) � � � ( � �� � ! � K �) � � * � + �� � � � H & �)5 (! � � � �:
� ���)5( � � �� � � � � � � � � �� � � ��
n = 2
X f ( x ) x f ( x ) 0 1 2
q2 2pq p2
0 2qp 2p2
���������� E ( x ) = 2 p ( q + p ) = 2 p
� % � � �n ; � ) � 3 ! 0 � �� % � � � 1 � � � ! � � � � � � * � + �� � � � ? � x �� � � � � �� 2 � � � ! 2 � 9 � % � F �8 3 � �� � �& = � � �)3 ( * 2 4 �� � � �) ��
! � ) � � � � ) �� � ( � �� � ! � � � 3 � ) � � � � � * � + �� � � � H & �)6 (! � � � � :
12 2
� ���)6( � � �� � � � � � � � � �� � � ��
n = 3
X f ( x ) X f ( x ) 0 1 2 3
q3 3p2q 3qp2
P3
0 3q2 p 6 q p2 3p3
���������� µ = E ( x ) = 3 p ( q 2 + 2 q p + p 2) = 3 p ( q + p )2 = 3 p
! � ) � � � � ) �� � � J � = � . � � � � � H � � �)* 4 �� � � �� ( � � � � � � ! 0 � " * � + �� � �� :
P � � � � H 6 n ; � ) � 1 2p � � � � H 6 n ; � ) � 2 3p� H 6 � � � n ; � ) � 3
0 � " * � + �� � � * 4 �� � J � � �� � � ( � � � � � � 7 H � / � % � � � � # � � � � � � � ! ; � ) � � � � � � �n p 2 � n� � � � � � � � .
�� � � � � � � � � � � � � � � � � � � � � � � � �p � � � � � � � � � � � � � � � q � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � ! � " � � # � $ � � � � � %n � # � $ � � �np=µ
12 3
! � � � * 4 �� � � � � % � / � � � ) � L ' � � @ � H � � � � � � � � � � � � )x( E =µ = ∑
=
n
0x(x) f x
= ∑=
n
0x
x-nxnx q p )( x
= ∑=
n
0x
x-nx q p! x)-(n !x !n x
= ∑=
n
1x1)-(x - 1)-(nx q p ! 1)]-(x - 1)-[(n ! 1)-(x
!n
= ∑=
n
1x
1)-(x - 1)-(n1-x! 1)]-(x - 1)-[(n ! 1)-(x
q p ! 1)-(n np
= 1)-(x-1)-(nn
1x1-x1n
1-x q P )( np∑=
−
= np (q+ p)n-1 = np
� � & �3 : ( � � � � ( % � 4 � � � � � � H 6100 ( 2 % 4 �� � � � � � � � � � � � �� � � F �
O F �8 3 � �� � �& = � � & & � � �: D � x 3 2 � �� � �2 & = � � � � � � �� � ! 0 � �� % � � � 1 � � � � � " � �
� ? � � H � � � F �8x� � � ! 0 � " * � + �� & � 2 � � � � P =
21
n = 100
124
np = � � � � � � ! 0 � " * � + �� � � * 4 �� �
µ = E (x) = np = 100
21 = 5 0
K �2 � � � � �2 2 � � o � � � � � � � � � � � � � � � � � � # �
� " � 9 � 6 F � �% � �)1 ( � � � ( % � 4 5 � � 6 ( � � � �� �n� � F � / � � � ) � � � � � � � �� 4∑ µ=σ f(x) )-(x 22
x � 2 % � � � n / � 2 � � � H 2 J � 1 E2 E3 � � � � � � ! 0 � " * � + �� � � � � � � � ( � � ( 1 � 8 9 � 6 � 8 � � � * � � � ) �.
� % � �n ; � ) � 1 ! 0 � �2 � % � � � 1 � � � ! � � � � � � * � + �� � � � �� � x � �� � � � ! � 9 � % � F �8 3 � �� � �& = � � � � � � �� �)1 ( � H 2 � * 4 �� �
� * � + �� � � � �� � � � ! � K ) � �)4 ( �� �p � � � � 2 � * � + �� � � � H � � � � � � � � � �� � ! � � � 3 � ) �)7 (! � � � � :
������)7( �� � � � � � � � � �� � � � �
n = 1
X f ( x ) x - µ ( x -µ )2 f ( x ) 0 1
q p
0-p 1-p
q2 q (1-p)2 p
�������� 2
xσ = p2 q + q 2p = pq ( p+ q ) = pq
125
� % � � �n ; � ) � 2 ! 0 �� 2 � % � � � 1 � � � ! � � � � � � * � + � � � � � ? � x
� � � � ! � 9 � % � F � 8 3 � � � � � & = � � � � � � � � �)2( �H 2 � * 4 � � � � � * � + � � � �2p � � � � ! � K ) � � � )5 ( � � � � 2 � * � + � � � � �H � � � � � � � �
� � ! � � � 3 � � 6 � � 8 � � � � �)8 (! � � � � : ������)8(
�� � � � � � � � � �� � � � � n = 2
( x -u) 2 f ( x ) x – u
f ( x ) X
4 p2 q2 (1-2p)2 2pq 4 (1-p)2 p2
-2p 1-2p 2-2p
q2 2pq p2
0 1 2
2xσ = 4 2q2p + ( 1 -2 p) 2 2 pq + 4 ( 1 -p) 2 p2
= 4 2q2p + ( 1 -4 p+ 4 p2 ) 2 pq + 4 q 2 p2
= 8 2q2p + 2 pq - 8 p2 q + 8 p2 q
= 8 p2 q ( q -1 + p) + 2 p q = 8 p2 q ( 0 ) + 2 p q = 2 pq
(�&�����
126
� % � � �n ; � ) � 3 ! 0 �� 2 � % � � � 1 � � � ! � � � � � � * � + � � � � � � x
� � � � ! � 9 � % � F � 8 3 � � � � � & = � � � � � � � � �)3 ( �H 2 � * 2 4 � � � � � � � ! � K ) � � � * � + � � � �)6 ( � �3p � � ( � � ) � � ( � � � � � I � � � � � � � � �
; � ) � * � + � � � � �H � � � � � 9 � 6 � 8 �3pq � � � � n ; � ) � 3 ; � ) � � � � � � � ! 0 � " * � + � � � � � � � � � � � � �H � � � :
pq � � � � n ; � ) � 1 2pq � � � � n ; � ) � 2 3P q � � � � n ; � ) � 3
% � � � � � � � ; � 2 ) � � � � � � � ! 0 � " * � + � � � � � � � � � � � � � � � ( � � � � � � 7 H � /� �npq 2 � n � � �
! � � � � � � � � �� * 4 � � � � � � � �� 4 /�� � ) � ' � � @ � H 3 � � � � � � � � � =σ2
x E (x2) – [ E (x)] 2
E (x2) = ∑=
n
0x
2 (x) f x
= ∑=
n
0x
x-nxnx
2 q p )( x
�� � � � �� � � � � � �� �� � � � # � � � � �� �p � � � �� �� � � �� � � �� q � � � � � � �� � � �� � � ��
&� �� �� � ) � � � � �� �� � � � # � � " � � # � $ �n �� � # � $ �
127
= ∑=
n
0x
x-nx2
! x)-(n ! x q p !n x
� � � $ � � % � � � �x2 2 � x(x-1)+ x = xnx
n
0xqp ! x)-(n !x
!n x] 1)-[x(x −
=
∑ +
= q p ! x)-(n !x ! xn qp ! x)-(n !x
!n 1)- x(x xnxn
0x
xnxn
0x
−
−
−
=
∑∑ +
= ∑=
n
2x ! 2)]-(x-2)-[(n ! 2)-(x
! 2)-(n 1)-n(n px q ( n-2) -( x-2) + E (x)
= n (n-1) p2 ∑=
n
2x ! 2)]-(x-2)-[(n ! 2)-(x
! 2)-(n px-2 q ( n-2) -( x-2) + np
= n (n-1) p2 2n)pq( −+ + np = n2 p2 – np2 + np 2xσ = E (x2) – [ E (x)] 2
= n2 p2 – np2 + np – n2 p2 = np – np2 = np (1 – p) = npq
� � �H � � � = npq2σ ! 2 % � � � � � H 2 � � � ; � 2 ) � � � � 2 � � � ! 0 � " * � + � � � ; � % � � � � � � � � ��
� � � � � K � � � � � npq 2 =σ=σ
128
�����4 : � " � 9 � 6 F � � % � �3 � � � ( % � 4 5 � � 6 � � � 10 0 � � � � � � � � � � 7 �
O F � 8 3 � � � � � & = � � � � � % � ; � % � � � � � � � � �� ���� : D �x 3 � � � � � & = � � � � � � � � � ! 0 �� � % � � � 1 � � � � � " � �
� � �H � � � 7 � 8x 3 � � � � � � � � ! 0 � " * � + � � & � P = 2
1
q = 21
n = 100 � � � � � � = 2
xσ = npq
= 100
21
21 = 25
; � % � � � � � � � � � = 25 npq ==σ = 5
� � � ����� � � � � � � � � � * � + � � /� � ) � D � 2 4 � 2 & � � � � � � � � � ! � � � K � � � � ! � � � ) �
F � � � � ( � � � + 7 � � � � . � � � � � � � L � � � 8 �� � � / � � � � � * ' � � C � � � �) /� � � � � � ( � ) � � � � ( � � 4 � � � � " � ...P � � ( F � � 2 � � ( � � � � � � 7 � � � 9 � � �
+ � � � � � �) ( � ) � � � � � � � � � � � " � ( � � � 2 � � � ! 0 � " * � + � � /� � ) � � � � � E
129
� 4 � � � � � � � ( � % � � � � � � � � � � � ! � / � � � � � * ' � � C � � � � D � � � � � � .� � ) �� � * � + � � � � � � � � ( " � � � � � :
� ( � " � � � + � � 3 � � � � � � � � = � � � � 9 � 6 Q � � � ! � � � � 0 ) � � � : R � ( � � � � � D � � ) � ! � � ) � � C � �� � � � �. R ( � ) � � ! � ( � � � � � � 1 � ! � � � � � � � � � � � �. R � + � � � � � � ! � � � % � � � � � � � �/� � � � ! � ( � � �. R ( � � 4 � � � ! � ! ) � � � � � � � � � � � � ! � � � ( � � � & � � � � � � � � � � � �.
� ( � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � :
R ( � ) � � � F � � � � S � 4 ( % � 4 ! � F � � � � � � � K � � % � � � � � R � 3 � � � � � & � @ ) ! � F � � � � � � � K � � % � � � � �� � � � � �
R T � 2 � 6 2 & � � � � ! 2 � � � ( � � 8 � � ! � ( � % � � � � � 5 � G � � � �� � � � � ) � �
/� 2 � � � � 2 � ( � � 4 H � * � � � � � 1 � � � � ! � � ) �� � � � � 1 � � � � � � �H � � � � � � 81 � � 2 � � ... ( 2 � � � � � � + 2 � � ! 2 � � � ( 8 � � ( � � � + F � � ! �
( � � � � � ( " . " � � � � � � � , � � � � � K � � � ( 8 � �: 1R � � ) � � � � � � � � � � � � � � � � + � � � � � � � � � � � � � � � � � ! � (
+ � � � � � � � � � 8 � � � � � � � � �. 2R � � F � 1 8 ( � � � � � � � � + � � � � F � 8 4 F � � � . � � � � � � � � � �
+ � � � � �H � ( � � � � � � F � � � � 7 H � � � � * � K ) � � � + � � � �.
13 0
3R ( � � � � � � F � 8 4 ( � � � + F � � � . " � � � � � � � � � � � � � �� � � L�� � L� � 1 8 � � � � F � 1 8 + � � � � � �3 � � � 6 � � � � C.
13 1
* 2 � + � � � ! � � 2 � ! 0 �� � % � � � � ) �� � � 1 � � � ! � � � � � � * � + � � � ��! � � � � � � % � � � K ) � ! � % � � � � ) �� � :
�����5: ! " � � � � # � $ % � # & � ' � � � ( # � � � � � � ) � � � � � # � * � � (
� � � # � � � � � � � � � # � � � � + � � � , � % � � � � - # � � � � � � " � � . � � � � � � /� � / � � ' � � � " % � � � # � $ % � � � � 0 � � # � � � �.
� � � � � � � � � � � � �� � �X � � � � � � � � � � � � � � � � � � � � � �� ! � " X
� �0 #1 #2 # ... � � ��� � % � � � � � � �&X � � � � � � ' ( ) � � � � � . � �� � � * �� ' + � � � :
f ( x ) = ! x e xλλ−
, � � x = 0 #1 #2 # ... e = . � �� / � � ��2. 7 18 28 λ = � 0 � � 1 � & 2 �3 � � � � ) ) ( + � � � + " � � � 4 � � �
� � � � � �5 � � ' � � � 0 ) ) � � � � � .
13 2
���� : D �x ! � � � � � � � � � � � � H � � � 0 � + � � � � � ! 0 �� � % � � � 1 � � � � � " � �
2 � � � � � � ( � � 4 � � � x ( � ) � � � ! � � ) �� � * � + � � λ ; � ) � 3 . /�� � 2 ) �� � ) �� � * � + � � ( 1 � 8 :
f (x) = ! x e xλλ−
p(x=4 ) = f (4 ) = ! 4)3( e 43−
= 24)81( 05.0
) C � �0 . 0 5 = (e-3 =
800135
= 16875.0
�����6: � � �� � � F � � � ( � � � + ( � � � ! � K � � % � � � � � � ) � � � � � �H 6
� � �20 ( � � � � � � � � � � � � � � * � � /� 4 3 × 10 /� 4 : i( K � � � 9 � ; � � � � �. i i( � 4 G � 9 � � � �� K � � 9 � ; � � � �.
13 3
���� : D �x ( � � � � � ! � K � � % � � � � � � " � � 3 × 10 2 � � � � � �x ! � 3 � � % � ! � � ) �� � * � + � �
1.5 2010 3 =×=λ
( � � � � � ! � K � � % � � � � � � ) � � � � � ; �3 × 10 � � 1. 5 � � ) �� � ( 1 � 8 , � � � � � � :
f (x) = ! x e xλλ−
i) P (x=0) = f (0) = ! 0(1.5) e 05.1−
= e-1. 5 =
0. 223 ii) p )� 4 G � 9 � � � �� K � �( = 1-p (x=0) =
1 - 0. 223 = 0. 7 7 7
� � � � � � � � � � � � �� � � � / � � � � � �H 6x� � ) �� � * � + � � & � ! � � � � � ! � � ) �� � � 1 � � � " � �
f (x) = ! x e xλλ−
x = 0, 1, 2, … λ > 0
! 0 �� � % � � � � ) �� � � 1 � � � � � � � * 4 � � � �x ; � * � + � � � � ( � % � ; � ) � ; � ) �λ� � ; � :
13 4
xµ = E (x) = λ 2
xσ = E (x2) – [ E (x)] 2 = λ � 2 2 � � � �:
E (x) = ∑∞
=0x(x) f x
= ∑∞ λ λ0
x-
! x e x
= ∑∞ λ
−λ
1
x-
! )1x( e
= ∑∞
λ−λλ
1
1-x-
! )1x( e
� � � � � � � �� � ' � � � � ! � � � % � � � � � � λe = 1 + !x ... ! 2 ! 1
x1
0x
2 λ=+λ+λ ∑=
∴ E (x) = λ=λ λλ e e -
E (x2) = )x(fx20x
∑∞
=
= ∑∞ λ λ0
x-2
! x e x
$ � � % � �X2 ( � � � � � X(X-1)+ XW � � �
135
= ∑∞ λ λ+0
x-
! x e x] 1)-(x[x
= ! x e x ! x
e 1)-(x[x x-
00
x- λ+λ λ∞∞ λ∑∑
= (x) E ! )2x( e x x-
2+
−λλ∞
∑
= λ+−λλ ∑
∞λ− ! )2x(
e 2-x
2
2
= λ 2 e-λ eλ + λ = λ 2 + λ
v a r (x) = E (x2) – [ E (x) ] 2 = (λ 2 + λ ) - λ 2 = λ
136
������� � �� � ��� � � �
1 ( 3 �� � , � � � 820 & � � � 0 � & � L � � 8 � 5 � � � ) E ( � � � 4 - � � 8 �
� � � � � K ) � � F � � X � * � ( � 0 �� � � ( � � � � : i (� K � % � � � �� � � 8 � 9 � � � 8 � �)� � �( )64/27(
ii ( K � % � � 4 G � 9 � � � �� � � 8 � 9 � � � 8 � � �)� � �( )256/175( 2 ( � � � � � � � � � � � � � � � �H 610 K � � � � K � � � � � � � � � � � � � ( 0 ) �
� � ! 0 �� � � � �Z ) ; � 9 �0. 2( � � � � � � � � � � � � � � � � : i () ; � 9 � K � � � � K � � � � � � � � � � � �Z ) 0. 1074(
ii ( " � � � � � �Z ) 9 � K � � � � � )0. 8926( iii ( 9 � K � � � � �3 � � � ( 0 ) � ) 0. 2( 3 ( � � �H 60. 15 � � % � 3 � � � � � � � � � + � � � � � � � � � � � T � � 6 Q � � 6 � �
� � � E15 5 . � % � � � � � 9 � 6 �+ & � . ( 2 � � � + & � � � � � � � � �H ? � � � � � � � � � � ( � � ) �:
i (K � % � � � � + � � ; � 3 � � � � � � � � � � � � � � � � � )0. 0873(
ii ( � � % � � 4 � � � �� � �� � � � � � � ) 0. 3185(
137
4 ( � � � � � � � � � � � �H 64 � � � � � � � k � & = � � � � � � � " � � 2 � ! � � � � � � * � + � � � � � � � � * � G � � � � � � 7 H � ! � ( � ) 3 � � � �k.
5 ( & � � � 7 ) �5 � �H ? � � � � � ; � 2 ) � �� 2 � � � � � � � � � � � � � � � � � � �
; � ) � � � � � � � � � � � � � � � �2/1 � � � � � � � � � E : i (I � � � � I � � � � � � � G � * � � � � � � � � � )16/1(
ii (F � � �� � � � � �� � �� �� � � � 4 G � 9 � F ) G � T � � � � � � � � )16/15( 6 ( ! 2 � % � ! 4 2 � � �� � � � � L�� � ! � % � & " " � � � � � H � � � ( � � �� �
� � + �H ? � E L ' � � �7 � � � � � K ) � J � ( � % � � 7 H � � � � � H � : i (5 � � � 7 � � ; � 9 � � 8 � � � � � )2187/128(
ii (5 � � � 7 � � �� 7 � � 9 � � 8 � � � � )2187/448( iii ( 5 ' � � 7 � � ; � 9 � � 8 � � � � � )2187/1( 7 ( � � � � � � � �� � �� � � � � ) � � � � � ! � � � �H 64 � �� � � � 1 � � � � � � �
( � 2 ) � �� 7 2 � � ( � � � � � ( � ) � � � � � � ( � � � � ! � ( � � � ( � � � � ! � � 2 � � � � 2 4 � � � @ � H ! � � � � ( � 0 �� � � / 4 � ( % ) � � �� 7 � � ( � � " � �
� � � � � : i (( � � 1 � � � M ( � � � � � � � � �� & � � * � � � � � )0. 075(
ii (� � 1 � � � � � �� � �� � � )0. 225( iii (� � " � � � � � � � " � � �( � � 1 � � ( � � � � � � � � �� & � � )0. 700(
138
8 ( 2 � & � � � � � � � K ) � 6 � � � I � � � � ( % � 4 � & � � � � 9 � � � � 8 �
� 8 � � � � � � % � � I � � 9 � L. 8 �. )256/63(
9 ( � � ( � � � � ! � ( ) 7 � K � . � � � � & � � � � � � � � � � �H 65/1 � � � �
9 � � � 8 � � � � � � � �4 � � � 4 G � 9 � � �� � � 253 � � . )0.766(
10 ( 3 � � � � �Z ) 9 � � �� � K � � � � � � � � � � ( � � � � � � � � � � � � � � � � ! �
� � ( � � � 83/1 � � � � 2 � I � 2 � � � �Z ) � � � � � � � � � � � � � � � � � ( � � � 8 ( � � 6 3 � � K � � � � �Z ))243/16.(
11 ( � � $ � � �0. 30 � � ( � 0 �� � � 3 � � � � � � � E � � � � � ( � � � � � � 6
� � � " � � � � � � � � � � � E A � � O � � � � � � 4 G � 9 � /& � � )0. 58(
12 ( ! 2 2 � � 2 2 � + � + � � � � � ! 2 � ( 2 2 � " � � ( 2 � � � 2 � � �H 60. 90 . 3 2 � � � � �10� D � 4 � � � � � � � � � E � � � � + � + � � �
)0. 6513(
13 ( � � % � � � /� 4 � � � �p , n 3 � ) � ; H � � � � � � � � ! 0 � " * � + � � � 9 3 � � � � � 5/18) 5/3 = p E 15 = n(
139
14 (� � � � � 3 � � ; H � � � � � � � � ! 0 � " * � + � � � � � � � � � * 4 � � � � �p = 0. 2 E
n = 35 (µ = 7 2σ = 5 . 6 )
15 ( � � � � Z � K � �232 3 � E ( � � 8 232 3 � + � � ! % � � � J � � ( � � 8 � � � � � � � K ) � � � 0 �� � � :
i (� � J � � � ' � � & � )e-1/2 = 0. 18 39( ii (� � J � � � � 4 � & � )2e-1 = 0. 7 35 6(
16 ( � � � * � 8 � ! � � � % � � C � �� � � � � � ) � � � � � �H 64 C � �� � � � � � � � � � � � & � :
i (� � % � & � ! � C � � ; � * � � � � � ) e-4 = 0. 018( ii (� � % � & � ! � " � G � 9 � C � �� � ( " . " * � � � � ) 0. 426(
17 (� � � � � �H 6 � � � � " � � � ( � � � � + � � � � � � ! � � � % � � � � � � � � � ) � � � � E � � ) �� � * � + � � * � � � � � % � � � � � � � � � � �H 6 � , 0 4 � I �
� � � � � :
14 0
i (, 0 4 � I � � . * � � � ; � � 8 � � � � ) e-2 = 0. 135( ii ( � � " � � � �4 � . �� 8 � � � % � � � 10, 0 4 � )0. 382(
18 (� � 9 � � � � � � � � � � � � � � . ( � � � � � � � � � * � � � � � � � ) � � � � � � � ) � � � 8 � ! � � � ( � � � & � � � � � � � � � � � � � � ) � � � � �H 6 � � � � � 4 �
O � � � � � 4 � � � ! � � � � � � � � � )0. 09(
19 ( T � � 6 & � � � � ! � � � ( � % � � � � � 5 � G � � � � � ) � � � � � �H 6 � � � � � � � ) � �4� � � � � � � � � E ( � � 8 � � ! � 5 � � :
i (� � � � 5 � � � � � � 3 � � 8 * � (e-4=0. 018 )
ii (� � � 8 � � � � � ( � � � ! � � 4 G � 9 � � � J � K � � � � � (0. 91)
20 ( � � � � � �� � � 7 � � � ( � � � + 3 � � � ! � K � � % � � � � � � ) � � � � � �H 6
3 � � � � � � � � � � � � � � ( % � � /�� 4 � F � �6 × 10/� 4 :R i (K � � � � � � � � � � � � � ) (e-6=0. 0025 )
ii (� � �� K � � � 4 G � 9 � & � � � � � ) (0. 997 5 )
21 ( ( � � � � � � ( � � G � ! � [ � � C � � � � � � ) � � � 3 � � � Q � � X * � 8 � ! � � � [ � � ) � � � � L � 0 �� � �200 ! � � � � � � � � � � � � K ) � � � �
3 � � � � � �500 � � . i ( " � G � 9 � � � � (0. 5 4 3)
ii ([ � � C . " � 4 G � 9 � (0. 4 5 7 )
14 1
(22C � � � � � % � � /� � � � K 8 � � � � , 0 4 3 � � 1 � ! � K � � � � � ! � � � ��2/2 � � % � � 3 � � 4 � � � � � � � � � � � � E 3 / ×7 /.
i (K � � % � � � � 3 � � )0. 00001( ii (� � �� K � � " � G � 9 � & � )0. 0001266(
14 2
������ ���� ��� � �� @
1( � 1 � � � � D �x( � � � ) � � � ( � � � � � - � � 8 � � � � � � 9 � � � � /� 4 � � � � �x ! � ( � � � � � � 0 E 1 E 2 E 3 E 4.
D �p H 0 � � � � � � � � 8 � K � ) � � � � � � " � � 41
205p ==
� 1 � � � �x F � � X � /� � K � ) � � � � @ � H � � � � � � � ! 0 � " � � � � � 9 � 6 � � � � � � 1 � � � � � � � � 8 � K � ) � � � � ��
; � ) � � � � "41
i( n = 4 x = 1
41p =
43q =
9 � � 8 � � � � � � � � ! 0 � " * � + � � � � ( 1 � 8 /�� � ) � � : P (x =1) = f (1) =
314
1 43
41
= 3
!!
!
43
41
3 14
= 4
6427
41
= 6427
14 3
ii ( P (x 1≥ ) = p (x =1) + p (x =2) + p (x = 3) + p (x = 4 ) = 1 – p (x =0) = 1 –
4
0
40
43
41
= 1 - 4
43
= 1 – 8 1/ 25 6 = 17 5 / 25 6
2( D �x( 0 ) G � � � � � " � � /� 4 � � � � � K � � � � & � � K � � � ! � � � x ! � ( � � � � � �10 E … E 1 E 0
�H 2 � � � n = 10
p = 0. 2
q = 0. 8 � � � � � � ! 0 � " * � + � � � � 3 1 � 8 /�� � ) � �
i) f (x) = ( )nx px qn-x p(x = 0) = f (0) = ( )10
0 (0. 2)0 (0. 8 )10 = (0. 8 )10 = 0. 107 4
14 4
ii) p(x ≥1) = f (1) + f(2) … + f(10) = 1 – f(0) = 1 – (0. 8 )10 = 0. 8 9 26
iii) p (x = 3) = f(3) = ( ) 73103 (0.8) (0.2)
= 120 (0. 008 ) (0. 8 )7 = 0. 2013 ≅ 0. 2
3( D �x� � F + & � G � � � � � " � � � � � % � 3 � � � � � � � 3 � � % � . / � 4 � � � � �x ! � ( � � � � � �15 E ... E2 E 1 E 0 0. 15 = p � � % � + & � � � � � � � � � � � � � � 0. 85 = q � � ) + & � � � � � � � � � � � � � �
15 = n
i) p (x = 0) = f(0) = ( )150 (0. 15 )0 (0. 8 5 )15
= (0. 8 5 )15 = 0. 08 7 3
ii) p ( x ≤ 1) = p (x = 0) + p (x =1) = f (0) + f(1)
145
= ( )150 (0. 15 )0 (0. 8 5 )15 + ( )15
1 (0. 15 )1 (0. 8 5 )14 = 0. 08 7 3 + 15 (0. 15 ) (0. 8 5 )14 = 0. 08 7 3 + 0. 2312 = 0. 318 5
4( / � � � � 2 � ( � � � � � �k ! � :
4 E 3 E 2 E 1 E 0 4 = n
61 = p 3 � ) 3 � � � � 9 � � � 8 � � � � � � � �
65= q
p (K = k) = f (k) = k4k4
K 65
61
−
� 1 � � � ! � � � � � � * � + � � � � ! � % � ! � � � � � � � � � � �K ����� ��� ���4� ��
K Pk Pk 0 (5 / 6)4 0. 48 23 1 4(1/ 6) (5 / 6)3 0. 38 5 8 2 6(1/ 6)2 (5 / 6)2 0. 115 7 3 4(1/ 6)3 (5 / 6) 0. 015 4 4 (1/ 6)4 0. 0008
146
5( D �x / � 4 � � � � � � � � G � � � � � � � � � G � � � � � " � � x ! � ( � � � � � �
5 E ... E1 E 0 5 = n 2
1 = p � � � � � � � � � � � � � � � � � � � � 2
1 = q � � � � � � � � � � � � � � � � � � � i)
� 4 G � 9 � 7 � � � � � � � � � � � � � � � � � 7 ) G � T � � � � � � � � � � � � � � � ; � E ( % � � � � 3 " . " � � � � � " � � � � � � & � � � � � � � � � � � � � � � �
& � � � � � � � �5 � � � � � � 5� � � . ; � ) � K � � � � � � � � � � � � � � � �
P = P(1) + P(2) + P(3) + P(4) = ( )( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )415
4235
3325
245
1 21 2
1 21 2
1 21 2
1 21 2
1 +++
= 1615
325
3210
3210
325
=+++
� � K � � � � � � � � � � � =1R) � � � � ( ) � � � � � � � � + ( ) � � � � � � � �
� � �( = 1 - [ ](0) P )5( P + = 1-
1615
321
321 =
+
ii) � � � � / & � � � � � � ; � ) � K � � � � � � � � � � � +� � � � �� � � / & �
147
P = P (0) + P (5 ) =
161
321
321
=+
6( D �x D + � � & � � � 8 � � ! � � � 5 � � � � � � � � � � � � � � � " � � 7 / � 4 � � � � � E � � H � x ! � ( � � � � � � 0 E 1 E 2 E ....7
D �p 5 � � � 7 � � 9 � � � 8 � � � � � � � � q5 � � � 7 � � 9 � � � 8 � � � / � � � � � � �
31 = p 32 = q
i) p (0) = f (0) = 707
0 32
31
= 0.05853 2187128
32 7
==
ii) p (1) = f (1) = 67
1 32
31
= 7 0.2048 2187448
32
31 6
==
iii) p (7 ) = f (7 ) = 077
7 32
31
= 0.0005 21871
31 7
==
7( D �x / � 4 � � � � � E 3 � � 1 � � � � � � � � & � � � � � � " � � x ! � ( � � � � � � 0 E 1 E 2 E ...9
148
9 = n 41 = p 43 = q
i ( � � � � � � � � � 3 � � 1 � � � M ( � � � � � � � � � � & � � * � � � � � � � � � �L� � 1 � � & � � � 8 � � � �
P (x = 0) = f (0) = 9
43
0
41 9
0
= 9
43
= 0. 07 5
ii (� � 1 � � � � � � � � � � � � � � � � � P (x = 1) = f (1) =
819
1 43
41
= 0. 225
iii (3 � � 1 � � " � � � � � � � " � � � � � � � � P (x ≥ 2) = p (x = 2) + p (x = 3) + … + p (x = 9 ) = 1- [ p (x =0) + p (x = 1)] = 1 – 0. 07 5 – 0. 225 = 0. 7
149
8( D �x � � � 7 � 8 � � 3 � � 9 � � � 8 � � � � � � � � � � " � � 3 � � 3 % � 4 5 � � �5 / � 4 � � � � � E � � � x ! � A � � � � ( � � � � � �
0، 1 E 2 ... 5 5 = n 21 = p = q
� � 9 � � . 8 � � � 8 9 � � & � � � � � 8 � � H 6 � 8 � � � � � � % � � I � �1 � � 2 � � 3 � � 4 � � 5 ; �
)5 � 5 ( � �)4 � 4 ( � �)3 � 3 ( � �)2 � 2 ( � �)1 � 1 ( � �)0 � 0( ; � ) � � � � � � 7 H � � � ( � � � � 9 � � � 8 � � � � � � � �
25
21 5
x
" � �L.
(0 , 0) = P(0) p(0)
= ( ) ( )
5050
50
21
21
21
21 5
0
= ( ) 2550 2
1
∴ � � � 8 � � � � � � % � � I � � 9 � . 8 � � � � � � � � � P = ( ) ( ) ( ) ( )
+
+
+
25
53
252
2551
25
21
21 5
21
21 5
0
150
+ ( ) ( )
21
21 5
4
2555
25
+
=
+
+
+
+
+
101010101010 21
2125
21 100
21100
2125
21
= 102252 =
1024252 =
25663
9( D �x � � 3 � � � � � � � � � � � G � � � � � " � � 25 / � 4 � � � � � E 3 � � x ! � ( � � � � � �0 E 1 E ...25
25 = n E 51 = p E
54 = q
9 � � � � � � ( % � � 9 � � � 8 � � � � � � � � � � � � � � � � 4 G � � � � " � � � � � � � � � ( % � � 9 � � � 8 � � �
P = (x = 4) + p (x = 5 ) + … + p (x = 25 ) = 1 – [ p (x = 0) + p (x = 1) + p (x = 2) + p (x =3)]
= 1 - ( ) ( ) ( )( )
+
+
+
223253
232252
24251
25250
54
51
54
51
54
51
54
51
= 1- (0. 0037 7 7 7 + 0. 0236110 + 0. 07 08 336 + 0. 135 7 644) = 0. 7 66
10( � � 3 � � K � � � � Z ) � � � � � I � � � � � Z ) � � � � � � � � � 2 � �
9 2 � � G � ( % � G � ( 0 ) G � 9 � K � � � � � � � � � � � ( � � � 8 ( � � 6
151
( � � � 8 ( � � 6 I � � � � � Z ) � � 9 � K � � � ( 0 � � � � ? �; � ) � �:
p = ( ) ( )
0111
4040 3
2 31
32
31
= 24316
316
31
32
5
4==
� �
P = 24316
31
32
32
32
32 =××××
11( D � x / � 4 � � � � � � � � � � � � � � � � " � � x! � ( � � � � � � :0 E 1 E
...6 � 2 � � � � � 2 � � � � � � � � � K � � � � � � � � � � � � � � � =p = 0. 30
� 2 � � � � � � � � � � � � � � � M K � � � � � � � � � � � � � � � =q = 0. 70
n = 6 � 2 � � � � � � � � � � � � 4 G � 9 � ( � � % � � � � � � � " � � � � � � � � � �
� � � � � � � � � � � � � �2 � � 3 � � 4 � � 5 � � 6 � � P (x ≥ 2) = 1- p (x < 2) = 1 – [ p (x = 0) + p(x = 1)]
� H 2 � � �
152
P = 1- [ ( ) ( ) 5161
6060 (0.7) (0.3) (0.7) (0.3) + ]
= 1 – ( 0 . 1 1 7 6 4 9 + 0 . 3 0 2 5 2 6 ) = 0 . 5 7 9 8 2 5
12( � 2 � � � H 6 � � � * 2 � � 9 2 � � 2 2 � � � � 2 � + � + � � � � � � � � 2 4 G � D �x / � 4 � � � � � 3 � � � � � � � � � + � + � � � � � � � � � 1 � � � � � " � �
x ! � ( � � � � � � )10 E ... E2 E 1 E 0(
� � � � � � � � � � = � D � 4 � � � � � � =p = 0. 1 � � � � � � � � � � = � D � 4 � / � � � � � � � =q = 0. 9
n = 10 � D � 4 � � � � � � =� � � D � 4 � / � � � � � � � A 4 � � � = � � � �
� � % � & � � � � � � � � � � � � A 4 � P = 1 – ( )10
0 ( 0 . 1 )0 ( 0 . 9 )10 = 1 – ( 0 . 9 )10 = 0 . 6 5 1 3
13( ; � ) � � � � � � � ! 0 � " * � + � � � * 4 � � � � � � ! � ) � � � � ) � � � np ) � � � � � � � ! 0 � " * � + � � � � � � � � � � ; �npq
n p = 9 … … … … … … … . ( 1 ) n p q =
518 … … … … … … ( 2 )
� � � � � �)1 ( �)2 (
153
9 q = 518
q = 52
4518
=
p = 1 - 53
52 =
( � � 4 $ � � % � �P ! � )1( n = 9 ÷ 15
345
53
==
14( 35 = n 0. 2 = p 0. 8 = q * 4 � � � � =np � � � � � � =npq
µ = np ∴ µ = 35 (0. 2) = 7
2σ = npq 2σ = 7 (0. 8 ) = 5 . 6
15 ( D �x 2 � � � � � � E F � � � � � � ( � � 8 � � ! � 5 � G � � � � � " � � x * � + � �
( � � 8 � � ! � � � � � J � � ) � � � � � � ) � � � 1
232232 ==λ
154
i( f ( x) =
!x e xλλ−
p ( x = 2 ) = f ( 2 ) = !2(1) e 21−
= e-1 / 2 = 0 . 1 8 3 9 C 2 � �
e-1 = 0 . 3 6 8 ii( P ( x < 2 ) = ∑
=
−1
0x
x1
!x(1) e
= e-1 ( 1 + 111 )
= 2 e-1 = 0 . 7 3 6
16 ( D �x � � ) � � � * � + � � & � � � � � � � � % � � � & � � � ! � C � � � � � � � � � � " � � 3 � � % � ; H � � =4
i( f ( 0 ) = !
04
0)4( e−
= e-4 = 0 . 0 1 8
ii( P (x ≤3) = ∑ λλ
!
x-
x)( e
155
= p (x = 0) + p (x = 1) + p (x = 2) + p (x = 3) f(0) = e -4 = 0. 018
f(1) = !
4
14 e− = 0. 07 2
f(2) = !2(4) e 24−
= 0. 144
f(3) = !3(4) e 34−
= 0. 19 2
f ( x ≤ 3) = 0. 426
17 ( D �x � � % � � � � � � � � ! � � � 1 � � � � ! � ( � � 4 � � � ! � � � % � � � � � � � � � ) � � � =
52 =λ
� � � ! � � � % � � � � � � � � � )5 , 0 4 � =2 ! � � � % � � � � � � � � � ) � � �10 , 0 4 � =4 ∴ 2 = λ, 0 4 � I � ! � 4 = λ, 0 4 � � � ! �
i) P (x = 0) = f(0) = !02 e 02−
= e -2 = 0. 135
ii) λ = 4 P (x > 4) = 1-P (x ≤4)
156
P (x ≤ 4) = ∑=
λ λ4
0x
x-
x! e
= P (x = 0) + P(x =1) + P(x =2) + P (x =3) + P(x =4) f(0) = e -4 = 0. 018 f(1) = e -4 (4) = 0. 07 2
f(2) = !2(4) e 24−
= 8 e -4 = 0. 144
f(3) = e -4
664 = 0. 19 2
f(4) = 664
!4(4) e 44
=
−
e -4 = 0. 19 2
0. 618 ∴ P (x > 4) = 1 – P ( x ≤ 4 ) = 1 – 0. 618 = 0. 38 2
18 ( D � x � � � � � 4 � � . � 8 � ! � � � ( � � � & � � � � � � � � � � � � � 1 � � � � � " � � 3 � % � � � � ) � � � * � + � � & � � � � � �2 = λ
P ( x = 4 ) = f (4) = 42 e 42−
= 32 e -2
= 3)135.0(2
=3
270
= 0. 09 0
157
19 ( D �x � � � � � ( � � 8 � � ! � ( � � � � � � 5 � G � � � � � 1 � � � � � " � � 2 �x 3 � % � � � � ) � � � * � + � � 4 =λ
i) P (x = 0) = f (0) = !02 e 42−
= e -4 = 0. 018
ii) P(x ≥ 2) = 1 – P (x < 2)
= 1 – [ P (X = 0) + (P(X =1)] = 1 – (0. 018 + 4 e -4) = 1 – (0. 018 + 0. 07 2) = 0. 9 1
20 ( D �x 3 � � � � � ! � K � � % � � � � � � " � � 6 × 10/ � 4 2 � � � � �x * � + � � 3 � � % � ; H � � � � ) � � �λ== 10
(10) (6) 6
x! e xλλ− f(x) =
i) P (x = 0) = f(0) = e -6 = 0. 0025
158
ii) P (x ≥ 1) = 1- P (x < 1) = 1-P (x = 0)
= 1 – f (0) = 1 - 0. 0025 = 0. 9 9 7 5
21( D � x � " � � 2 � � ! � [ � � � � � � �500 � � ∴ λ==
200500 5.2
!x e )x(f
x- λ=λ
i) P ( x ≤ 2 ) = f ( 0 ) + f ( 1 ) + f ( 2 ) f ( 0 ) = e -2 . 5 f ( 1 ) = e -2 . 5
f ( 2 ) = 2.5-25.2e 3.126 2
(2.5) e=
−
P ( x ≤ 2 ) = 6 . 6 2 5 e -2 . 5 = ( 6 . 6 2 5 ) ( 0 . 0 8 2 ) = 0 . 5 4 3
ii) P ( x ≥ 3) = 1 – P ( x < 3) = 1 - { }f(2) f(1) )0(f ++
= 1 – 0 . 5 4 3 = 0 . 4 5 7
159
22 ( D �x ( � � 4 � � ! � K� � % � � � � � � " � � 3 × 7 /
λ==λ=λ
221 11.5 x!
e )x(fx-
i) P ( x = 0 ) = f ( 0 ) = e -1 1 . 5 = 0 . 0 0 0 0 1
ii) P ( x ≤ 1 ) = f ( 0 ) + f ( 1 ) f ( 1 ) = e -1 1 . 5 1 1 . 5
f ( 0 ) + f ( 1 ) = 1 2 . 5 e -1 1 . 5 = 0 . 0 0 0 1 2 6 6
16 0
������ ����
16 1
������� ���� � � � � �� � � � � � ��
The Normal Distribution
2 � " � �� ( 2 � � � � � � � � % � + � � � � / � � � � ! % � � � � � * � + � � � � � � % � � � 2 � � � � � ! 2 � 7 � � 8 � � * ' � � � � � � 3 � � � � E , . � � � 9 � � � % � ) �
� � � � 4 � E 5 8 � � � � � % � + � � � � � � � � " � � � � � 3 � ) � ,"( � % � � � � � " H 2 J � 3 � � � � � � � � � % � + � � � � / = % � � ? � @ � H � E 3 � � � � 4 . � �) � � � � � � % � + � � �
� + � � � � ( � � � � � � � � 5 � � � % � + � � �) / � 2 � � � � 3 � � � � � � � / � � � � � � � , � � � �7 � � � � � � ( � 2 � 2 � " � ! 2 � * � + � � � � � H � / � � ) � � E 3 � � � � 4 . � � H J �
� � K � � � � � � � � � ! � 3 % ) � � � � � � � ) � 3 � � 7 � � � � � � � � � � � ( � � � 8 � � M � 3 � � % � � � � % � + � � � F � � � � � � � � % � � � $ � � � �.
� � � � ; � / � % � � * � + � � � � � H � � � � � � � � � � � � � �D e M o i v er / �1733 I � M / � % � � 7 � % � � � � G au s s / � ! � 1809 � H 2 � � % � �
I � 2 M * 2 � + � � ; � 3 � ) � L ' � � * � + � � � �G au s s D i s t r i bu t i o n � H 2 & � � ( 2 8 ( 2 � � � � 2 2 � � � � � � � � � � � � � � � ( � ' � � � 3 8 � � * � + � � � �
� � � � � � ! 0 � " * � + � � � 2 " � T � � % � + � � �. � 2 � / � � � � � � � � E ! % � � � � � 9 � � � � � � 9 � � � * � + � � � � � H � 9 � � � �
2 � 1 � � � � � 2 � � � � ( 2 � " � ( 2 � � � 9 � � � � � 8 � � � � J � ; = � * � + � � 9 � � � � 3 2 � ; H � � ! 0 � � � % � � � � � � " � � � ! % � � � � � * � + � � � � 9 � � � � � E * � + � � � � � H �
� 2 � . 2 � � 2 " � � � � K� ) � ; � ) � ; H � � ! � ) � � � � ) � � � � � ! ) � � � � � � � � � � � ) � � � . � � 9 � 6 7 � � � � � � � 7 � � � � 3 � 4 3 � � � � � � ! ) � 4 � � � �
16 2
� ) � � � � � � ( � & �) � 2 � & � � � � ! 2 � � G � � � � � � � � 7 � � K � � � � 3 % � � � � � � ( @ � H * � � - � � 8 � � � � � � � � ; � ) � 9 � � � � � � � � � ( � ) � � � � �
2 � 1 � � ; � � 2 � � � � ( 2 � " � 3 � � � 9 � � � � � � � ( � ) � � � ! � � � � � � � � � � � 8 � � ! 0 � � � �.
� � � � � �1 9 � � � � � � � " � � :
� � �1 � � � � � � � � � �
� 2 � � 2 � � & � � � � ( � % � � � � � � � � � � � � � � � ! 0 & � � � � � @ � � � 2 � ( � � 4 K) � $ % � � � & ' % � ! � 2 ) � � � � 2 ) � � � � 2 �) * 2 4 � � � � (µ
; � % � � � � � � � � � �σ � � 2 � � � � ! 2 � ( 2 � % � � � � � � � � � , � � � � 4 � E � � � � � ! � � � � � ! � ) � � � � ) � � � ! � � � � & � � � � ; � % � � �2 � 4 � � E
� � � ! � � � � � ; � % � � � � � � � � � � � � � ! � ) � � � � ) � � � , � � � � �3
16 3
µ = 2 µ = 4 µ = 6 ���2
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
5.0=σ 1=σ
2=σ µ = 3
� � �3 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � 1( ! � ) � � � � ) � � � � � � � " � � � ! % � � � � � 9 � � � � � �µ 9 � � � � � � � � ; �
3 � � � � 9 � 9 � � � � � � � " � � ! � ) � � � � ) � � � ) � 9 �. 2( 9 � 6 9 � � � � � � � � � 3 � ) � � � / ) � � � � 8 � � � � � ! � ! � ) � � � � ) � � �
� � � � ) � � � � � ) 4. 3( � � � � ; � ) � 9 � � � � � � � � � 3 � ) � � � D � � � �. 4( ( � � 4 � � � � & �µ � σ� 9 � � � � � � ? � ( � � � � � A � � � � ! % � � � � :
� R ! � � � �68. 26 % � � � � � � � � � � � * � � 3 � ) � � � � �σ R µ � σ+ µ
x
x
f(x)
16 4
68.26
K R ! � � � � 95. 46 % � � � � � � � � � � � * � � 3 � ) � � � � �σ2Rµ � σ2+ µ
Q R ! � � � � 99. 74 % � � � � � � � � � � � * � � ( � ) � � � � �σ3 Rµ � σ3 +µ
� R ! � � � � 95 % � � � � � � � � � � � * � � ( � ) � � � � �σ1. 96Rµ � σ1. 96 +µ
2 �R ! � � � � 99 % � � � � � � � � � � � * � � ( � ) � � � � �σ2. 58Rµ � σ2. 58 +µ
� - ' � � � � � � � � � � � !4! � � � � :
� 8 � � � � ! 0 � � � % � � � 1 � � � � � �x 3 � � � � � � � H 6 � % � � � D + � � 3 � � ( � � � � � ( 1 � 8 � � 7 � % � 3 � � � � � � � ( � " � :
x
%13.6
68.26%
%34.13
%34.13
%13.6
2.14% %2.1 4
9 9 .7 4%
95.46%
6 8 .26
16 5
f (x) =
2 -x 21
e 21
σµ−
σ π
C � � ∞ < x > ∞− µ ! % � � � � � � 1 � � � ! � ) � � � � ) � � � ! � e 3 � � � 4 � � " 2. 71828 π ; � ) � � ( � � � � � � � ( � ) � � � ! � 3. 14159 ! � 2 ) � � � 3 � ) � ; H � � ! % � � � � � * � + � � � � K� � � �µ 3 2 � � � � � 2σ
) � 3 � � � � � � � � � � ; � ) � ; � % � �σ ( + � � � K� � �)2σ , µ(N . " 2 � � )2 0,4 (N ! � ) � � ) � 3 � � % � � � � � 1 � � ! � % � )* 4 � � ( ; � ) �20 � � � � �
; � ) �4) ; � ) � � % � � � � � � ; �2.( � � � �f (x) 2 ) � � � D � 2 � � � � � � � � � � � � ( � " � 3 � � � � 2 � � ( �
. 8 � � � 0 � � � � L� � 1 � � � � � � � � � � - � � 8 � � � � � � � ) � 3 � � � � � 7 H � 9 � � � � 3 � ) �µ3 � � � � � 2σ � " � � � � � � � � � � � � � 4 � � � * � � b , a � � )b < x <
a (p 3 � � � � � 9 � � � � � � � 7 � 8 � � � � ( � ) � � � ; � ) � � f (x) � � � � � � � � � x = b , x = a
µ a b
f(x)
16 6
� � � �5 � � � � � � � � � � � � � � � � � � � � � � � � � � � The Standard Normal
D i s tri b u ti on � 1 � � � � 9 � ! ) � � � � � � ; � % � � � ! % � � � � � � 1 � � � � / ) � , � � � 1 � � � � 9 � ; � � � � � 3 � � � � � � 8 ! � ) � � � 3 � ) � ; H � � ! % � � � � �
N(0.1) + � � � 3 � + � � � z! � 3 � � � � � ( � " � 3 � � � � � � � � f (x) = 2/2z-e
21π
∞− < z < ∞
� � � � �f (z) 9 � � � � � � � ( � ) � � � D � � � � � � � � � � � ( � " � 3 � � � f (z) ; � ) �1
2/z-
-
2e
21 ∫
∞
∞π
d z = 1
� � � � � � � � � � � ; � % � � � ! % � � � � � ! 0 � � � % � � � 1 � � � � D � 4z � � � � � � � � � � � z = a Ez = b � 2 � � � � � ( � " � 3 � � � 9 � � � � � � � 7 � 8 � � � � ( � ) � � � � � f (z) � � � � � � � � � � � � � � � � � � z = a E z = b; � ) � ; �
P (a < z < b ) = ∫b
a
π21 2/z2e− d z
( � ) � � � � � ) � � � �)� � � � � � � � � ( ! � � ; � % � � � ! % � � � � � 9 � � � � � � � � � � � � � � � ! � 7 � % �2, � � � � .
� � � � � � � � � � � � � � � 4 � � � � 8 � � � � ! % � � � � � � 1 � � � � D � 4 � � � � � � K ) � � � 2 � � � � � ( 2 � " � 3 � � � 9 � � � � � � � 7 � 8 � � � � ( � ) � � � K) � � � �f (x)
2 � � � � � � � � � � � � � � � � � � �X � � � � � �f(x) � ) � � � � � � � � � % � � � ' � �
167
! � ) � � �µ � � � � � � � 2σ 7 � � � � � � M � � 4 � H J � � � � & � � � � � � � � � � 4 L ' � � � � H J � � � 3 � � � � ! � � � � � � 8 � � � 1 � � ! 0 � � � % � � � 1 � � � � � 6�
) � � � � � � � � M � � 3 � ? � T � � ; � � � � 7 � � � � � � M 3 % 4 � � � � � � ) � � � K � � � � � 3 � � � / � � � ! % � � � � � 9 � � � � � � � � �µ ,2σ , x * � � � 2 ) � � � � � �
! 2 % � � � � � � 1 � � � � � � � � � � ( � � % 8 � � 7 H � 9 � K 1 � � �)2σ, µ (N 9 2 � 6 ; � % � � 1 � �N(0.1) � � � � ! � � 7 � � � � � � � 3 � ) � � � K ) � � � � �
� � � � � � ! � � � ) � � � 7 � % �2, � � � � .
������� ����� ����) 2σ, µ (N �� � �� �� ������� ����� � �� ). 10(N
! 0 � � � % � � � 1 � � � � � � � H 6x ! % � � � � � * � + � � � � 3 � )2σ, µ (N � ? � ! 0 � � � % � � � 1 � � � �
σµ−= xz ; 2 � % � � � ! 2 % � � � � � * 2 � + � � � � 3 2 � � � � �
N(0.1) 3 � � � � � � 8 ! � ) � � � 3 � ) � � � % � � % � � � L % � + � � 3 � � � � � ; � 1 ! � ) � � � 3 � ) � ! % � � � � 1 � � ; � � � � µ 3 � � � � � 2σ 9 � 6 3 � � � � � � � �
3 � � � � ; � % � ! % � � � � 1 � �σµ−= xz ! 2 % � � � � � * � + � � � � � � � � � � H & � �
7 � � � � � � � 7 � 8 � � � � ( � ) � � � K ) � � � � � ; � % � ! % � � � * � + � � 9 � 6 � � � � � ! � � �6.
� � � � H 6b / � 4 � � ( � � % � ( � � 4 x � � P (x ≤ b) = P (x - µ ≤ b - µ )
168
= P
σµ−≤σ
µ− bx
= P
σµ−≤ bz
µ b
f(x)
x
169
Q z = σµ− b
�����6 � � � �� �� � )2σ, µ (N � �� z
! � 9 � � � � � � � � � 3 = � � � � M ( � ) � � � P (X ≤ b) = P (Z ≤
σµ - b )
' � � � � � � � P (X > b) = 1 – P (X ≤ b)
= 1 – P (Z ≤ σµ - b )
� � � � H 6�b< a) � � � ! � � �7 ( � � P (a ≤ X ≤ b) = P (Z ≤
σµ - b ) - P ( Z ≤
σµ - a )
) � 9 � 6 ( � ) � � � ; � ) � ; �b ) � 9 � 6 ( � ) � � � A 4 � a
f(z)
z
170
µ a b �� ��7
� � � � � � 8 � � � � � � � � � � � ( � " � 3 � � � 9 � � � � � � " � � � � ( � 8 � � � i ) ∫
∞−
xf(z)d z = ∫
∞−
0f(z) d z + ∫
x
o
f(z) d z
= 21 + ∫x
o
f(z) d z
i i ) ∫−
∞−
xf(z) d z = ∫
∞
x
f(z) d z
= 21 - ∫x
o
f(z) d z
� 2 � � � � � � � � � � � � � ) � � � / � � � ( � � ) � � � � � � � � � � � ! � ! � � � � @ � H � � / � � � ( � � � � � � � � � � � � K ) � � ; � % � � � ! % � � � � � 9 � � � � � �x 3 2 � � � � � �
/ � � � � � � � � � � Q � � � ) � � � � � & � � �x( � � ) � � .
x
171
������� : � � � � �1 :
� � � � H 6X * � + � � � � 3 � ! % � � � ! 0 � � � � � 1 � � )4 E 3 (N 2 � � * � � � � � � � � � � �X � � � � � � � � � � � 3 E 5O
� � ���: ; � % � * � + � � 9 � 6 * � + � � � � � � � �
z = σµ−x
z1 = 0233=
−
z2 = 1235=
− p (3 < x < 5) = p (0 < z < 1)
� � � 3 � ) � � � ; � ) � �z = 0 � z = 1� ! � � � ! � � ! � � � � � � � �
0 1 � � � � � � � � ! � �2 ; � ) � , � � � � 0. 3413
172
� � � �2: ! 0 � � � % � � � 1 � � � � � � H 6X ! % � � � � � * � + � � � � 3 � )2,σµ (N
� � � �:
i) P ( )x σ+µ<<σ−µ ii) P ( )2x2 σ+µ<<σ−µ iii) P ( )3x3 σ+µ<<σ−µ
� � ��� : i) z1 = 1)( −=
σµ−σ−µ
z2 = 1)( =σ
µ−σ+µ
∴ P ( )1z1(P)x <<−=σ+µ<<σ−µ � � � ( � ) � � � � � F � � � � �z = -1 � z = 1
= � � � ( � ) � � �z = 1 � z = 0 + ( � ) � � � � � �z = 0 � z = 1
-1 0 1 ! � ) � � � � ) � � � ) � � � � � � 9 � 9 � � � � � � � " � � � ( � � � � �0=µ � �
� � � ( � ) � � � z = -1 �z = 0 � � � ( � ) � � � ; � ) � z = 0 � z = 1 � � � � � � � � � � ; � ) � � 7 . � � � � � � � ! �0. 3413
Z
173
∴ P ( σ+µ<<σ−µ x ) = 0. 34 13 + 0. 34 13 = 0. 6 8 26 = 6 8 . 26 %
� � ! � % � � H � �68. 26 % � � ' * � � 9 � � � � � � � � � ( � ) � � � D � � � � � �! � ) � � � � ) � � � � � � � � � ; � % � � � � � �.
ii) P ( )2 x 2 σ+µ<<σ−µ z1 = 2 )2 ( −=
σµ−σ−µ
z2 = 2)2( =σ
µ−σ+µ
∴ P ( )2z2(P)2x2 <<−=σ+µ<<σ−µ � � � 7 � 8 � � � � ( � ) � � � ; � ) � �z = -2 � z = 0 + F � 8 � � � � ( � ) � � �
� � �z =0 � z =2 � � � � � � � � � = 0. 4 7 7 3 + 0. 4 7 7 3 = 0. 9 5 4 6 = 9 5 . 4 6 %
� � ; �95. 46 % � D � � � � � � � � � � � � � � � ' * � � 9 � � � � � � � � � ( � ) � � � 8 ! � ) � � � � ) � � � � � � � � � % �.
Z
174
-2 2 iii) z1 = 3)3( −=
σµ−σ−µ
z1 = 3)3( =σ
µ−σ−µ
∴ P ( )3z3(P)3x3 <<−=σ+µ<<σ−µ
� � � 7 � 8 � � � � ( � ) � � � ; � ) � �z = -3 � z = 0 + F � 8 � � � � ( � ) � � � � � �z =0 � z = 3� � � � � � � � � :
= 0. 4 9 8 7 + 0. 4 9 8 7 = 0. 9 9 7 4 = 9 9 . 7 4 %
� � ; �99. 74 % � � 2 ' * 2 � � 9 2 � � � � � � � 2 � � ( � ) � � � D � � � � � �3
! � ) � � � � ) � � � � � 3 � � % � � � � � � �.
-3 3
%99.74
175
� �� � �� � � �� � �� �� ������� � �� ����
1. � � �( � � � � � � � � � � � � � � � � ; � % � � � ! % � � � � � * � + � � �:
i) (0. 7 < z< 0 (P )0. 2580( ii) (1. 2 < z< 0 (P )0. 3849( iii) (0 < z< 1R (P
)0. 3413( iv) (0. 96 < z< 0. 82R (P
)0. 6254( v) (R0. 6 < z (P )0. 2742( vi) (1. 1R > z ( P )0. 8643(
2. � � � ( � ) � � � � � � �0. 5R = z
3. ! 0 � � � % � � � 1 � � � � � � � H 6x ! % � � � � � * � + � � � � 3 � N (2, 1) � � � � :
i ) (4 > x (P )0. 0227( ii ) (2 < x < 0 (P )0. 4773(
4. 2 & � � 2 � % � � � 3 2 � + � � � % � � � � T � � � K. � � � � � � � � � � H 6
! � ) � � � 3 � ) � ; H � � ! % � � � � � * � + � � � �68. 50 3 2 � � � � � � 3 8 � � ; � % � � �2. 33 8 � � :
176
i( � � + � ( % � � � � � � K� � ; � � � � � � � � � � � � � � � �6 / � � 4 � )723 8 � � ) (0. 0643.(
ii( � � � / & � � � � � � � � � � � � H � � ( % � � � � ! � 3 � � � � ( � ) � ! � �70 � 72 3 8 � � )0. 1935(
5. � � � � � � � � � H 63000 � 2 � � ! % � � � � � * � + � � � � � � � H � � K� �
! � ) � � � � ) � � � ; � 2 ) � � � � 2 � G � 7 H & � 170 � � 2 � � � � � / 2 ) ; � ) � & � ; � % � � �5 / ) :
i( � � " � � / & � � � � � � � H � � ( � � � � ( � ) � � � � �185/ ). ii( � � / & � � � � � � � + � � � H � � ( � � � � � � � � � � �185/ ). iii( � 2 ) � � � � � , � 2 � / & � � � � � � � H � � ( � � � � ( � ) � � � � � H 6
� � � � � � � % � � � � � � � 4 ��x � � 0. 2881 � 2 � � � O� � � � � � H �.
6. 6 ! 0 � � � % � � � 1 � � � � � � � Hx 3 � 2 ) � ; H 2 � � ! 2 % � � � � � * � + � � � � 3 �
! � ) � � �80 ; � % � � � 3 � � � � � � 0. 30� � � J � : i( )80. 36 ≤ X (P ii( � � % � �C � � � H 6 0. 95) = C ≤ X ( P
7. & � � � � � 4 � � � � � � $ � � � � G � � � � � � W � � � * � 8 �4
1 * � + � � & � * � 8 � � � � H � Q � � � � � � G � � 4 � � � � � � 7 � � � � / )
! � ) � � � 3 � ) � ! % � � �2. 5 ; � % � � � 3 � � � � � � / �0. 0025 E / 2 �
177
� � 2 � 2 � � 4 � � � 2 � � ! 2 � � � � a � ( � � 0 � � � ( � ) � � � ! � � �2. 4951 � 2. 5049O/ �
8. � � % � � � � � ) � � � � � � H 64 ; � % � � � � � � ( � ) � � ! � � � � �
0. 50 ( � 2 ) � ! � � � � % � � � % � + � � * � � � � � G � � � � � H 6� � � � � � � � � � � � � ' � � � � � H � � � � % � �2. 5 � 3O( � ) � � ! � � � � �
9. ! 0 � & � � � � � 8 � � � � � � ) � � � � � � H 61500 � � � � 2 � ( � )
; � % �50 � � � � � � � � � � % � � � % � + � � * � � � � ( � ) : i( � � 4 , � � � ) � � 8 � � �1400( � ) . ii( � � 6 � � " � � S � % � ) � � 81550( � ) . iii( � � � � S � % � ) � � 8 � � 61450 � 1550( � ) .
10. ; � ) � 3 � � 3 � � � � � � + � � � � � � � � , � � � 8 10 I 2 � � �
(O U NCE S ) ; � % � � � � � � 1. 5 3 � + � � � � � � � � � + � �� I � � � � � 2 � & � + � * � � ! � � � � � � � � � ( � ) � ! � � � � % � � �7. 9 � 12. 4
OI � � �
11. 6 � � � � � � � � H 6 * 2 � + � � � � 2 & � ( � ) � � ! � � � � � � T � � 3 � ) � ; H � � ! % � � � � �100 ; 2 � % � � � 3 2 � � � � � � � � � � � � � 5
� � � � � � . � : i( � � � � + � ) ( � � � � � � � � � � � � � � � � � � 100 � � � � � � � . ii( � � � � � ) ( � � � � � � � � � � � � � � � � � � 100 � � � � � � � .
178
iii( � � 2 � � � � � ) ( � � � � � � � � � � � � � � � � � � 100 � 110 � � � � � � �.
179
� �� � �� � � �� � �� �� � )������� � �� ����(
1 . � � � � � � � � ( � � � � � � � � � � � � � � � �27 � � � , � � � � . 2 . ! 2 2 � ( 2 2 � � � � � � ( � 2 2 ) � � �
� � 2 � � � ! 2 � ( 2 = � � � ( � ) � � � � � 2 � F � 8 � � � � �∞R � 0. 5R
� 2 � ( � 2 ) � � � � " � � ! � �0.5 9 � �∞ � 2 � ( � 2 ) � � � A 4 �
9 � � � 80. 5 � � � � � � � � � = 0. 3085 = 0. 1915 – 0. 5
3 .; � % � * � + � � 9 � 6 ! % � � � � � * � + � � � � � � � � i(
σµ= -z
z 2
12-4 z ==
∴ P (X > 4 ) = P (z > 2) = 0. 5 – 0. 4 7 7 3 = 0. 0227
ii ( 2- 12 - 0 Z1 ==
-0 . 5 Z 0 2
18 0
0 12 - 2 Z2 ==
∴ P (0 < X < 2) = P (z > 2) = P (-2 < z < 0) = P (0 < z <2)
= 0. 4 7 7 3
4 . i.
Z = σµ - x
= 2.53.5
2.368.5 - 72
=
= 1. 5 2 ∴ p (x > 7 2) = p (z > 1. 5 2) = 0. 5 -0. 4 35 7 = 06 4 3
ii. z1 = 2.3
68.5 - 70 = 0. 6 5
z2 = 2.368.5 - 72 = 1. 5 2
∴ P (7 0 < x < 7 2) = P (0. 6 5 < z < 1. 5 2)
0
18 1
= 0. 4 35 7 – 0. 24 22 = 0. 19 35
5 . . i z = 3
5170185
=−
∴ P (x > 18 5 ) = P ( z > 3 ) = 0. 5 – 0. 4 9 8 7 = 0. 0013
ii. 3000 = % 100 x = % 0. 13 x = 100.0
(3000) )0013.0(
= 3. 9 ≅ 4
iii. z =
5170 - x
( � � 4 � � � � � � � � � � � z 7 = � � � � ( � ) � �0. 2881 ! � 0.8
∴ 0. 8 = 5170 - χ
x = 174
0 3
x 0
0 . 28 8 1
182
6. i.
Z = 1.2 3.0
80 - 36 - 80=
)2.1 z ( p ) 80.36 x ( p ≤=≤∴ = 0. 5 + 0. 38 49 = 0. 8 8 49
ii. z =
3.080 c −
( 2 � � 4 � � � 2 � � � � � � � � � �z ( � 2 2 ) � � ( 2 2 � � � � �)0. 50 –
0. 95 ( ( � ) � � ; �0. 45 ! � 1. 64
∴ 64.13.080 c
=−
∴ c = 8 0. 49 2
444 3444 21
������� � ��� =0.95 � �� �� ������ �� � �0.4 5 � �� � � � �� �� � � �� � �
� � � � � �� � �∞ � � ! " �� ∞− � � � ∞ 7.
z1 = 96.10025.0
5.24951.2=
−
z2 = 96.10025.0
5.25049.2=
−
∴ P (2. 49 51 < x < 2. 5049 ) = P (-1. 9 6 < z < 1. 9 6 ) = 2 (0. 4750)
1 .2
0 0
0 . 4 5
-1. 9 6 1. 9 6
183
= 0. 9 5
8. z1 = 3-
5.045.2=
−
z2 = 2- 5.043=
−
∴ p (2. 5 < x <3) = p (-3 < z <-2) = p (2 < z < 3) = 0. 49 8 7 – 0. 4773 = 0. 0214
9. i. z =
501500 - 1400 = -2
∴ p (x < 1400) = p (z < -2 ) = 0. 5 – 0. 4773
= 0. 0227 ii .
z = 501500 -1550 = 1
∴ p ( x > 1550) = p (z > 1 ) = 0. 5 – 0. 3413
-3 -2
-2 0
0 1
184
= 0. 158 7
iii. 1z =
501500 - 1450 = -1
2z = 1 50
1500 - 1550=
∴ p (1450 < x < 1550) = p (-1 < z 1< ) = 2 (0. 3413) = 0. 6 8 26
10 .
1.4 1.5
10 - 7.9 z1 ==
1.6 1.5
10 - 12.4 z2 ==
∴ p (7. 9 < x < 12. 4 ) = p (-1. 4 < z < 1. 6 ) = 0. 419 2 + 0. 4452 = 0. 8 6 44
11. i.
-1 0 1
-1. 4 0 1. 6
185
z = 0
5100 - 100
=
p ( x > 100 ) = p (z > 0 ) = 0. 5
ii. z = 0
5100 - 100
=
p ( x < 100 ) = p (z > 0 ) = 0. 5 iii.
0 5100 - 100 z1 ==
2 5100 - 100 z2 ==
∴ p (100< x < 110) = p (0 < z < 2) = 0. 4773
186
��� ���� ���� � �� ��� �
1R & � ( 0 � � � � ! � ( � 0 � � � % � � ( � � � � � � � � �3 � � � 2 ) � � � 2 � � � � � � G � 5 � Z � F � � � � � � ) ) � � I � � � � K ) � . � � � 2 � � � � K ) � �
� � � � � ( 0 % � � � � � . )8/3(
2R � � � H 6A1 E A2 ! 2 � � % � � c � � � � ! � � � " � � S � 2 � � 0. 5 = P(A1) � 0. 7 = P(A2)
�0. 3 = P(A1 A2):
i ( � �A1 � A2 � � � � � � " � � ) � � � � ; � ) � � � & � � � � � � D � � � � � � �(
ii (� � � �)A2 � � A1 ( P )0. 9( iii ( � � � �)A1 R A2 ( P )0. 2( iv ( � � � �)A2 R A1 ( P
)� � � � ) �( 3R � � � H 6A � B ! 2 � � % � � c � � � � ! � � � " � � S � 2 � � 0. 4 =
P(A) � 0. 7 = P(B) �0. 3 = P(AB) � � � � � K ) � � :
i ( � � " � � � � � � � D � 4 �A � � B )0. 8(
187
ii ( D � 4 �A� � B � & � � I � � � )0. 5(
iii ( D � 4 � / � �A )0. 6( iv ( D � 4 �A D � 4 � / � � � B )0. 1(
4R ( � � � � � C � � � � � � � � � � � � � � � � 7 � � � � 7 � � � � � � ! � � � :
i (A1 � � � = � � � � � 4 � � D � � � � 7 )36/6( ii (A2 � � " � � � � � = � � � � � 4 � � D � � � � 10
)36/3( iii (A3 � 4 � � D � � � � ; � 2 ) � � � � 2 � � 2 4 � � � � 2 = � � � �5
)36/10( iv (A4 � � % � � 2� 4 G � 9 � 7 � � � � 7 � & =
)36/11(
5R ( � � � � � � � � � � � K ) � � , � ) � � � � � � � � I � � ! � :
i ( � � G � � � � � � � $ � 9 � ! � � + D � � � � � �6 ) 6/3( ii ( ! � " � � � � � �6! � � + D � � � � � � $ � 9 � )18/3( iii (� + D � � � � � � � � � 4 � D � � � � � � � � � � � � H 6 ! �5 ) 6/4( iv ( ! � " � � � � � �6 � � G � � � $ � 9 � 4 ) 6/1(
v ( ; � � � � G � � � � � � � � � � � H 6 ; � � D � � � � � � ) 18/9(
188
6R � � ( 8 � � � � � H � � � + M 9 � 6 % � � & � � � 4 � � � � � � � 8 3 � � � ! � � � + 1 � � ( � 8 6 ! � � & � �
31 � K ) � J � � � + 2 1 � � ( � 8 6 � � � � .
)9/5( 7R � � � � H 6A P ) � � 20 % � � � � � & % � � � � � � � �50 % � 2 �
P ) � � � � � � 5 � � � � � � & � � � B � 2 � � � ! 4 � � � 90 % � 2 � 5 � � � � � � & � � � . � 2 & % � � � � 2 � � � � K � K � ) � H ? �
5 � � � � � � K � � � � � � � � � � � � � � K ) � � . )0. 82( 8R � H 6 , � ) � � � � Z ) � � ! � 5 2 � � 3 � � � � K � � ) � � � K � � � � �
� � � � � � � � � � � � � � �A O 3 � � � � ! � � � ! � )0. 5555=18/10(
9R 5 8 � � � F � � ! � K � � � � � � � � � � � � � � � H 6 =0. 40 � � � � / � � � � � � :
i ( � � � � � � � � � K � � - � � � � �5K . � ) 0. 2592( ii (� 4 G � 9 � � � � � K � � - � � � � � ) 0. 92224( iii ( - � � � � �4 " � G � 9 � K . � ) 0. 98976( iv (� � � - � � � � � � ) 0. 07776(
10R D � � � � 9 � � 8 � � � � � � � � � � � �5 ! � � � � � � C . " � � ! � � +6 O � � �
)0. 17342(
189
11R � � & � [ � 8 � � ( � 8 6 � � � � � � � � H 60. 3 � � 2 � � � � 6 � � � � � � � � � ! � & 4 . � 6 / + . � � P � � � 8 � � 9 � � � & � � ( � 8
� 4 G �80.%
� � � � � � : J � � � � � � � � =0. 7 d � � P � � � 8 � � � � � � � =n)0. 7( � � K � � � � �0. 8> 1-(0. 7)n
0. 2R > (0. 7)n 0. 2 < (0. 7)n
(0. 7)1 = 0. 7 , (0. 7)2 = 0. 49
(0. 7)3 = 0. 743 , (0. 7)4 = 0. 2401 (0. 7)5 = 0. 16 8 07
� � � P � � � 8 � � 5 = n ∴
19 0
������������
19 1
� � � � � � )1( � � � � � � � � � � � � � � � � � � � � e-x
e-x x e-x x e-x x e-x x e-x x
0.00034 8 .0 0.002 5 6.0 0.01 8 4.0 0.1 35 2 .0 1 .000 0.0
0.00030 8 .1 0.002 2 6.1 0.01 7 4.1 0.1 2 2 2 .1 0.9 05 0.1
0.0002 8 8 .2 0.002 0 6.2 0.01 5 4.2 0.1 1 1 2 .2 0.8 1 9 0.2
0.0002 5 8 .3 0.001 8 6.3 0.01 4 4.3 0.1 00 2 .3 0.7 41 0.3
0.0002 3 8 .4 0.001 7 6.4 0.01 2 4.4 0.09 1 2 .4 0.67 0 0.4
0.0002 0 8 .5 0.001 5 6.5 0.01 1 4.5 0.08 2 2 .5 0.607 0.5
0.0001 8 8 .6 0.001 4 6.6 0.01 0 4.6 0.07 4 2 .6 0.549 0.6
0.0001 7 8 .7 0.001 2 6.7 0.009 4.7 0.067 2 .7 0.49 7 0.7
0.0001 5 8 .8 0.001 1 6.8 0.008 4.8 0.061 2 .8 0.449 0.8
0.0001 4 8 .9 0.001 0 6.9 0.007 4.9 0.055 2 .9 0.407 0.9
0.0001 2 9 .0 0.0009 7 .0 0.0067 5.0 0.050 3.0 0.368 1 .0
0.0001 1 9 .1 0.0008 7 .1 0.0061 5.1 0.045 3.1 0.333 1 .1
0.0001 0 9 .2 0.0007 7 .2 0.0055 5.2 0.041 3.2 0.301 1 .2
0.00009 9 .3 0.0007 7 .3 0.0050 5.3 0.037 3.3 0.2 7 3 1 .3
0.00008 9 .4 0.0006 7 .4 0.0045 5.4 0.033 3.4 0.2 47 1 .4
0.00008 9 .5 0.00055 7 .5 0.0041 5.5 0.030 3.5 0.2 2 3 1 .5
0.00007 9 .6 0.00050 7 .6 0.0037 5.6 0.02 7 3.6 0.2 02 1 .6
0.00006 9 .7 0.00045 7 .7 0.0033 5.7 0.02 5 3.7 0.1 8 3 1 .7
0.00006 9 .8 0.00041 7 .8 0.0030 5.8 0.02 2 3.8 0.1 65 1 .8
0.00005 9 .9 0.00037 7 .9 0.002 7 5.9 0.02 0 3.9 0.1 50 1 .9
19 2
/ 2 4 , 2 � �)2 / (! % � � � � � 9 � � � � � � � � � � � ) � � � )A � � � � � � � � � � � � � � � � � � 0 , 2(
A Z A Z A Z A Z
0.4 207 1 .4 1 0.326 4 0.9 4 0.1 8 08 0.4 7 0.0000 0.00 .4 222 1 .4 2 .328 9 .9 5 .1 8 4 4 .4 8 .004 0 .01 .4 236 1 .4 3 .331 5 .9 6 .1 8 7 9 .4 9 .008 0 .02 .4 25 1 1 .4 4 .334 0 .9 7 .1 9 1 5 .5 0 .01 20 .03 .4 26 5 1 .4 5 .336 5 .9 8 .1 9 5 0 .5 1 .01 6 0 .04
.4 27 9 1 .4 6 .338 9 .9 9 .1 9 8 5 .5 2 .01 9 9 .05 .4 29 2 1 .4 7 .34 1 3 1 .00 .201 9 .5 3 .0239 .06 .4 306 1 .4 8 .34 38 1 .01 .205 4 .5 4 .027 9 .07 .4 31 9 1 .4 9 .34 6 1 1 .02 .208 8 .5 5 .031 9 .08 .4 332 1 .5 0 .34 8 5 1 .03 .21 23 .5 6 .035 9 .09
.4 34 5 1 .5 1 .35 08 1 .04 .21 5 7 .5 7 .039 8 .1 0 .4 35 7 1 .5 2 .35 31 1 .05 .21 9 0 .5 8 .04 38 .1 1 .4 37 0 1 .5 3 .35 5 4 1 .06 .2224 .5 9 .04 7 8 .1 2 .4 38 2 1 .5 6 .35 7 7 1 .07 .225 8 .6 0 .05 1 7 .1 3 .4 39 4 1 .5 5 .35 9 9 1 .08 .229 1 .6 1 .05 5 7 .1 4
.4 4 06 1 .5 6 .36 21 1 .09 .2324 .6 2 .05 9 6 .1 5 .4 4 1 8 1 .5 7 .36 4 3 1 .1 0 .235 7 .6 3 .06 36 .1 6 .4 4 30 1 .5 8 .36 6 5 1 .1 1 .238 9 .6 4 .06 7 5 .1 7 .4 4 4 1 1 .5 9 .36 8 6 1 .1 2 .24 22 .6 5 .07 1 4 .1 8 .4 4 5 2 1 .6 0 .37 08 1 .1 3 .24 5 4 .6 6 .07 5 4 .1 9
.4 4 6 3 1 .6 1 .37 29 1 .1 4 .24 8 6 .6 7 .07 9 3 .20 .4 4 7 4 1 .6 2 .37 4 9 1 .1 5 .25 1 8 .6 8 .08 32 .21 .4 4 8 5 1 .6 3 .37 7 0 1 .1 6 .25 4 9 .6 9 .08 7 1 .22 .4 4 9 5 1 .6 4 .37 9 0 1 .1 7 .25 8 0 .7 0 .09 1 0 .23 .4 5 05 1 .6 5 .38 1 0 1 .1 8 .26 1 2 .7 1 .09 4 8 .24
.4 5 1 5 1 .6 6 .38 30 1 .1 9 .26 4 2 .7 2 .09 8 7 .25 .4 5 25 1 .6 7 .38 4 9 1 .20 .26 7 3 .7 3 .1 026 .26 .4 5 35 1 .6 8 .38 6 9 1 .21 .27 04 .7 4 .1 06 4 .27 .4 5 4 5 1 .6 9 .38 8 8 1 .22 .27 34 .7 5 .1 1 03 .28 .4 5 5 4 1 .7 0 .39 07 1 .23 .27 6 4 .7 6 .1 1 4 1 .29
.4 5 6 4 1 .7 1 .39 25 1 .24 .27 9 4 .7 7 .1 1 7 9 30 .4 5 7 3 1 .7 2 .39 4 4 1 .25 28 23 .7 8 .1 21 7 .31 .4 8 2 1 .7 3 .39 6 2 1 .26 .28 5 2 .7 9 .1 25 5 .32 .4 5 9 1 1 .7 4 .39 8 0 1 .27 .28 8 1 .8 0 .1 29 3 .33
0 z
19 3
.4 5 9 9 1 .7 5 .39 9 7 1 .28 .29 1 0 .8 1 .1 331 .34
4 6 08 1 .7 6 .4 01 5 1 .29 .29 39 .8 2 .1 36 8 .35 .4 6 1 6 1 .7 7 .4 032 1 .30 .29 6 7 .8 3 .1 4 06 .36 .4 6 25 1 .7 8 .4 04 9 1 .31 .29 9 6 .8 4 .1 4 4 3 .37 .4 6 33 1 .7 9 .4 06 6 1 .32 .3023 .8 5 .1 4 8 0 .38 .4 6 4 1 1 .8 0 .4 08 2 1 .1 33 .305 1 .8 6 .1 5 1 7 .39
.4 6 4 9 1 .8 1 .4 09 9 1 .34 .307 9 .8 7 .1 5 5 4 .4 0 .4 6 5 6 1 .8 2 .4 1 1 5 1 .35 .31 06 .8 8 .1 5 9 1 .4 1 .4 6 6 4 1 .8 3 .4 1 31 1 .36 .31 33 .8 9 .1 6 28 .4 2 .4 6 7 1 1 .8 4 .4 1 4 7 1 .37 .31 5 9 .9 0 .1 6 6 4 .4 3 .4 6 7 8 1 .8 5 .4 1 6 2 1 .38 .31 8 6 .9 1 .1 7 00 .4 4 .4 6 8 6 1 .8 6 .4 1 7 7 1 .39 .321 2 .9 2 .1 7 36 .4 5 .4 6 9 3 1 .8 7 .4 1 9 2 1 .4 0 .3238 .9 3 .1 7 7 2 .4 6
. . .� � � � � � � � A Z A Z A Z A Z
0.4 9 9 7 3.4 7 0.4 9 8 4 2.9 4 0.4 9 20 2.4 1 0.4 7 00 1 .8 8 .4 9 9 8 3.4 8 .4 9 8 4 2.9 5 .4 9 22 2.4 2 .4 7 06 1 .8 9 .4 9 9 8 .4 9 .4 9 8 5 2.9 6 .4 9 25 2.4 3 .4 7 1 3 1 .9 0
.4 9 9 8 3.5 0 .4 9 8 5 2.9 7 .4 9 27 2.4 4 .4 7 1 9 1 .9 1 .4 9 9 8 3.5 1 .4 9 8 6 2.9 8 .4 9 29 2.4 5 .4 7 26 1 .9 2 .4 9 9 8 3.5 2 .4 9 8 6 2.9 9 .4 9 31 2.4 6 .4 7 32 1 .9 3 .4 9 9 8 3.5 3 .4 9 8 7 3.00 .4 9 32 2.4 7 .4 7 38 1 .9 4 .4 9 9 8 3.5 4 .4 9 8 7 3.1 .4 9 34 2.4 8 .4 7 4 4 1 .9 5
.4 9 9 8 3.5 5 .4 9 8 7 3.2 .4 9 36 2.4 9 .4 7 5 0 1 .9 6 .4 9 9 8 3.5 6 .4 9 8 8 3.3 .4 9 38 2.5 0 .4 7 5 6 1 .9 7 .4 9 9 8 3.5 7 .4 9 8 8 3.4 .4 9 4 0 2.5 1 .4 7 6 2 1 .9 8 .4 9 9 8 3.5 8 .4 9 8 9 3.5 .4 9 4 1 2.5 2 .4 7 6 7 1 .9 9 .4 9 9 8 3.5 9 .4 9 8 9 3.6 .4 9 4 3 2.5 3 .4 7 7 3 2.00
.4 9 9 9 3.6 0 .4 9 8 9 3.7 .4 9 4 5 2.5 4 .4 7 7 8 2.01 .4 9 9 9 3.6 1 .4 9 9 0 3.8 .4 9 4 6 2.5 5 .4 7 8 3 2.02 .4 9 9 9 3.6 2 .4 9 9 0 3.9 .4 9 4 8 2.5 6 .4 7 8 8 2.03 .4 9 9 9 3.6 3 .4 9 9 0 3.1 0 .4 9 4 9 2.5 7 .4 7 9 3 2.04 .4 9 9 9 3.6 4 .4 9 9 1 3.1 1 .4 9 5 1 2.5 8 .4 7 9 8 2.05
.4 9 9 9 3.6 5 .4 9 9 1 3.1 2 .4 9 5 2 2.5 9 .4 8 03 2.06 .4 9 9 9 3.6 6 .4 9 9 1 3.1 3 .4 9 5 3 2.6 0 .4 8 08 2.07 .4 9 9 9 3.6 7 .4 9 9 2 3.1 4 .4 9 5 5 2.6 1 .4 8 1 2 2.08 .4 9 9 9 3.6 8 .4 9 9 2 3.1 5 .4 9 5 6 2.6 2 .4 8 1 7 2.09 .4 9 9 9 3.6 9 .4 9 9 2 3.1 6 .4 9 5 7 2.6 3 .4 8 21 2.1 0
.4 9 9 9 3.7 0 .4 9 9 2 3.1 7 .4 9 5 9 2.6 4 .4 8 26 2.1 1 .4 9 9 9 3.7 1 .4 9 9 3 3.1 8 .4 9 6 0 2.6 5 .4 8 30 2.1 2 .4 9 9 9 3.7 2 .4 9 9 3 3.1 9 .4 9 6 1 2.6 6 .4 8 34 2.1 3 .4 9 9 9 3.7 3 .4 9 9 3 3.20 .4 9 6 2 2.6 7 .4 8 38 2.1 4
19 4
.4 9 9 9 3.7 4 .4 9 9 3 3.21 .4 9 6 3 2.6 8 .4 8 4 2 2.1 5
.4 9 9 9 3.7 5 .4 9 9 4 3.22 .4 9 6 4 2.6 9 .4 8 4 6 2.1 6 .4 9 9 9 3.7 6 .4 9 9 4 3.23 .4 9 6 5 2.7 0 .4 8 5 0 2.1 7 .4 9 9 9 3.7 7 .4 9 9 4 3.24 .4 9 6 6 2.7 1 .4 8 5 4 2.1 8 .4 9 9 9 3.7 8 .4 9 9 4 3.25 .4 9 6 7 2.7 2 .4 8 5 7 2.1 9 .4 9 9 9 3.7 9 .4 9 9 4 3.26 .4 9 6 8 2.7 3 .4 8 6 1 2.20
.4 9 9 9 3.8 0 .4 9 9 5 2.27 .4 9 6 9 2.7 4 .4 8 6 5 2.21 .4 9 9 9 3.8 1 .4 9 9 5 3.28 .4 9 7 0 2.7 5 .4 8 6 8 2.22 .4 9 9 9 3.8 2 .4 9 9 5 3.29 .4 9 7 1 2.7 6 .4 8 7 1 2.23 .4 9 9 9 3.8 3 .4 9 9 5 3.30 .4 9 7 2 2.7 7 .4 8 7 5 2.24 .4 9 9 9 3.8 4 .4 9 9 5 3.31 .4 9 7 3 2.7 8 .4 8 7 8 2.25
.4 9 9 9 3.8 5 .4 9 9 6 3.32 .4 9 7 4 2.7 9 .4 8 8 1 2.26 .4 9 9 9 3.8 6 .4 9 9 6 3.33 .4 9 7 4 2.8 0 .4 8 8 4 2.27 .5 000 3.8 7 .4 9 9 6 3.34 .4 9 7 5 2.8 1 .4 8 8 7 2.28 .5 000 3.8 8 .4 9 9 6 3.35 .4 9 7 6 2.8 2 .4 8 9 0 2.29 .5 000 3.8 9 .4 9 9 6 3.36 .4 9 7 7 2.8 3 .4 8 9 3 2.30
.4 9 9 6 3.37 .4 9 7 7 2.8 4 .4 8 9 6 2.31 .4 9 9 6 3.38 .4 9 7 8 2.8 5 .4 8 9 8 2.32 .4 9 9 7 3.39 .4 9 7 9 2.8 6 .4 9 01 2.33 .4 9 9 7 3.4 0 .4 9 8 0 2.8 7 .4 9 04 2.34 .4 9 9 7 3.4 1 .4 8 0 2.8 8 .4 9 06 2.35 .4 9 9 7 3.4 2 .4 9 8 1 2.8 9 .4 9 09 2.36 .4 9 9 7 3.4 3 .4 9 8 1 2.9 0 .4 9 1 1 2.37 .4 9 9 7 3.4 4 .4 9 8 2 2.9 1 .4 9 1 3 2.38 .4 9 9 7 3.4 5 .4 9 8 3 2.9 2 .4 9 1 6 2.39 .4 9 9 7 3.4 6 .4 9 8 3 2.9 3 .4 9 1 8 2.4 0
19 5
��� ���� ����� �� ��� �
19 6
��� ���� ���� � �� ��� �
1R & � ( 0 � � � � ! � ( � 0 � � � % � � ( � � � � � � � � �3 � � � 2 ) � � � 2 � � � F � � � � � � ) ) � � I � � � � K ) � � � � G � 5 � Z � . � � � 2 � � � � K ) � �
� � � � � ( 0 % � � � � � . )8/3(
2R � � � H 6A1 E A2 ! 2 � � % � � c � � � � ! � � � " � � S 2 � � � 0. 5 = P(A1) � 0. 7 = P(A2)
�0. 3 = P(A1 A2):
i ( � �A1 � A2 � � � � � � " � � ) � � � � ; � ) � � � & � � � � � � D � � � � � � �(
ii ( � � � �)A2 � � A1 ( P )0. 9( iii ( � � � �)A1 R A2 ( P )0. 2( iv ( � � � �)A2 R A1 ( P
)� � � � ) �( 3R � � � H 6A � B � � � " � � ! 2 � � % � � c � � � � !S � 2 � � 0. 4 =
P(A) � 0. 7 = P(B) �0. 3 = P(AB) � � � � � K ) � � :
i ( � � " � � � � � � � D � 4 �A � � B )0. 8(
19 7
ii ( D � 4 �A � � B � & � � I � � � )0. 5(
iii ( D � 4 � / � �A )0. 6( iv ( D � 4 �A D � 4 � / � � � B )0. 1(
4R � � � � � � � � � 7 � � � � 7 � � � � � � ! � � � ( � � � � � C � � � � � � � :
i (A1 � � � = � � � � � 4 � � D � � � � 7 )36/6( ii (A2 � � " � � � � � = � � � � � 4 � � D � � � � 10
)36/3( iii (A3 ; � 2 ) � � � � 2 � � 2 4 � � � � 2 = � � � � � 4 � � D � � � � 5
)36/10( iv (A4 � � % � � 2� 4 G � 9 � 7 � � � � 7 � & =
)36/11(
5R � � � K ) � � , � ) � � � � � � � � I � � ! �( � � � � � � � � :
i ( � � G � � � � � � � $ � 9 � ! � � + D � � � � � �6 ) 6/3( ii ( ! � " � � � � � �6! � � + D � � � � � � $ � 9 � )18/3( iii ( � � � 4 � D � � � � � � � � � � � � H 6 ! � � + D � � � � � �5 ) 6/4( iv ( ! � " � � � � � �6 � � G � � � $ � 9 � 4 ) 6/1(
v ( � � � � � � H 6 ; � � D � � � � � � ; � � � � G � � � � � ) 18/9(
19 8
6R � � ( 8 � � � � � H � � � + M 9 � 6 % � � & � � � 4 � � � � � � � 8 3 � � � ! � � � + 1 � � ( � 8 6 ! � � & � �
31 � � + 2 1 � � ( � 8 6 � � � � � K ) � J � .
)9/5( 7R � � � � H 6A P ) � � 20 % � � � � � & % � � � � � � � �50 % � 2 �
P ) � � � � � � 5 � � � � � � & � � � B ! 4 � � � � 2 � � �90 % � 2 � 5 � � � � � � & � � � . � 2 & % � � � � 2 � � � � K � K � ) � H ? �
5 � � � � � � K � � � � � � � � � � � � � � K ) � � . )0. 82(
8R 5 2 � � 3 � � � � K � � ) � � � K � � � � � � H 6 , � ) � � � � Z ) � � ! � � � � � � � � � � � � � � � �A O 3 � � � � ! � � � ! �
)0. 5555=18/10( 9R � � � � � � � � � � � � H 6 5 8 � � � F � � ! � K � � � =0. 40 � � � � / �
� � � � � :
i ( � � � � � � � � � K � � - � � � � �5K . � ) 0. 2592( ii (� 4 G � 9 � � � � � K � � - � � � � � ) 0. 92224( iii ( - � � � � �4 " � G � 9 � K . � ) 0. 98976( iv (� � � - � � � � � � ) 0. 07776(
10R D � � � � 9 � � 8 � � � � � � � � � � � �5 . " ! � � � � � � C � � ! � � +6 O � � � )0. 17342(
19 9
11R � � & � [ � 8 � � ( � 8 6 � � � � � � � � H 60. 3 � � 2 � � � � 9 2 � � � & � � ( � 8 6 � � � � � � � � � ! � & 4 . � 6 / + . � � P � � � 8 � �
� 4 G �80.%
� � � � � � : J � � � � � � � � =0. 7 d � � P � � � 8 � � � � � � � =n)0. 7( � � K � � � � �0. 8> 1-(0. 7)n
0. 2R > (0. 7)n 0. 2 < (0. 7)n
(0. 7)1 = 0. 7 , (0. 7)2 = 0. 49
(0. 7)3 = 0. 743 , (0. 7)4 = 0. 2401 (0. 7)5 = 0. 16 8 07
P � � � 8 � � � � �5 = n ∴