# דף נוסחאות הסתברות

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: 2) = 2 ( : )( : 1. ) + = () .2 + = 2 ( 3. ) V(X)=0 .4 () ( X" : = = = [ = ] : = = = = : " , .A =n' , ) - ( A n ( )A = , n , - A - )P(A : " - 0 1 : -X' . , + 1, , 1, )~(, 1 1+ ) ( n

, : ) = + ( : k A n : 1. ' k .2 ,A -} {1..k .3 ,A k - n :A

! ! nk

)1 + = 1 (

: 1. ' k - .A 2.' } {1k- .3 A k - n .n k

=

= =

=

+ 2

=

1 2)1+ ( 21

!n : ! k! nk

: 2 , . : n ", X ,P ' : ) 1,2, . . , | ~(, = =

: 1. k .2 ,n k .3 ,n k n = )( )(

:

1

) = = (1

: " : ,P . X ' : ) 1,2 , ~(

: )P = = () (| ' : 1 , 2 . . , - Ai = 1=i

= = 1

1

=

1

=

1 2

= P = P )( )(

) > = (1 k: ) > + > ) = ( > : -': N D , n, X' )X~HG(N, D, n= =

=

)(|)( : )(

) ( ) =1 (

: : 1 , 2 . .

= P Ai B

)P(A i B )P(B

=

) P(B|A i )P(A i ) n P(B|A )P(A j j 1=j

: ' . ': . 2 ' , . ' ' . . ' ". ~ 0,1, . . = = !

=

) 1( =

1

" :2. )P = (| 1. )P A B = P(A) P(B 3. )P = (| B,A" , B A, B , B . A B , - 0, " " - = = 1 = () :A

= = * -

" : , . : x,y" k l ,x :y ) - = , = = ( = ) ( = . 0 ,

" , ': 1. n " : ) P Ai Aj = P(Ai ) P(Aj

2. n " : ": ) o (x ) P Ai1 Ai2 . . . Aik = P Ai1 P Ai2 P(Aik

) , = ( ()( () = ( = + 2 , + + + = + + + 2[ , ] , + , 0=) Cov(x,y x,y . " " Cov, ", , . . 0 " ) :Cov(x,y ":

= = =

)(= ,= )(=

:

+ ) 2V(X)=E[(X )2 ]= 1 2 = 1 + 2 2 ( = "

: ": E = = ) = ( : , . : )E X + Y = E X + E(Y 2 ' E aX + b = aE X + b :a,b : )E Y X = k = lY l P(Y = l|X = k : X" ) g(x' X: (..2 )x1,x X + 2 E g X = g x1 P X = x1 + g x 2 P X = x -

2) = 2 ( = 1

=

: )( " :

,

, 0

) ~(, ' : ) ~(, :

. ' X:

+ = E X: 2 2 = X: 21

. . : 1,1- x,y ) (,

=

0

, < , , >

" : ) ~( ' ": 0 ex , x = ,0 ~exp: )( 1 . X: = E . ' X 0 1 ex , x = ,0 1 2 =

1

1 1 ,

(V

) =1

x1..xn (") : : : ) + = : ,X 2,1: : , 1:)(=1+= )(+= )(=11+= )(+=

=

=1

) = ()(

) + = ) = 1P = 1 + = + 2( = 2| + =

. X:

= P = 1 + =

=

) ( + = ' t" : " () 0 1 .

) > + > ) = ( >

: : n: ) = ( ) = ( n: n :COV 2 =)V(X : 1. , - X . 2. X = : X = + 3. )1 = ( ) E( 4. ) E( 5. " 6. 2 Cov, Cov-2+ .i,i , 0=Cov " 0 X x f 1 . ' = ( ) -X : . = : . ( ) . 1 = lim Fx t = 0 lim Fx t . : ) = ( , :

: 1. , 2. ' 0

=

1 2 2

2) ( 2 2

" X ' , 2 : ) 2 ,X~N(, - , : 0 = 1 = 2 : )1,0( X~N: 898.0 = 72.1 1.27 = = < ) = 1 (z 1 . 1 < < = 2 > = 1 < = 1 : , : ) = ( P 1-P : 70.1 = 7758.0 1.07 , . 5.0, : = : ) 2 X~N(, ,

=

:

X z , -z 071 071 571 > 175 = > ) 25 ,071(= P(z > 1) :X~N 5 5 : = + :

: () () :

"

: 5 7 = : 3 : 2 2 . !3 : 1: 6 3 ( 63 '), : A 3 9 2 3

)( = )( < )( >

)5.0 + ( 0.5 < < )5.0 + ( )5.0 ( < )5.0 ( )5.0 + ( >

, ':

3 1

6 3

3 2

+

3 1

+

3 2

6 : 3

1 . 2 .

2 , 3

3: 4 , 8 . : . . . 2 (, ' ').2 : : 501 = 2 !4 2 , !4 3 , ( , , ). !4 , . 8 6 4

2 =

.

1

. : 1 2 3 4 , , 4, 3 , !4. 2 2 . = 9. 4: , 2 , 2 2 . 2 ", : B A C -D , .6 A 1: 2 1 .

501

. =

-2 ,

4 2

4 2

. :

:

2: : 5 6 2 ( ', '), ( ', '). B : {,}, {,}, 6 {,} 3 23

C 3 , : ) 1 P(A 3 4 6 D : 2 4 - 4 , 4 2 , . 5: 21 , : B k : !01 ) 1 + 2 - 2! (12 k , !01 , 1 + (12 k k .

: 1. 2. N, D, 1 , ? N 3. 4. , , , . 5. 6. -Cov 7. : 8. - 9. , 2 E : 2 = 2 10. !!!!!!!!!!!!!!!!!! 11. , 010,100

=

1: n, . . X' . . :X : : n . 1+ : . X 0 2- .n : n 2: - 4 , 3 , 2 , 1 . . X' . "

2

1 . 9.0 . 2 ? . , . . 5 ? . , - 5 ? . " -2 " - " 7 " ?

. X: : : , y 01 . Y 1 9. : 2 10 k , : 10-k: 01 = = 2

: : B -1 2 . , ) P(B|A 2 = = 1, A )P(B|A)P(A )P(B 21

= P AB

:

9381.0 = !2 1P B A = e1

. : : , ? . . ' : : , 2 3 . y 3,2,1,0 : 2 2 . )1+5( =) - p(y=k k, : 5 1+ ,k )1+ 5-(k . , .!2!9 !012

5

= ) 2( + P 2 P = !2 )P(B|A)P(A )P(B

1 0.9

5 + 0.1 9.09381.0 371.0

25 !2

371.0 =

= P AB

=

: 1 : = e2

': -C 0 ,P X

= C2

01 !0 1e

: : .

+

1 1 1e

: : ~ 5, D -2 ' )2 > = 0.1 0 + 0.9 ( 0 . .

3: , - " . ? ( ) : A 1 = L

}2 { , 1, ) 1 , ( 7.0 = 7.0 60.0 0=

': 5 :512.0 = 5) (e

2

=

() , : )512.0(~

L + G 1 )2.0 3.0( 1=

60.01 7.0 =

1

': B 7 A -2

98980.0 = 0 1.0 + 2 = P A = 0.9 P X , : 7 = = 1 5, 7 = = 5 5,

5 P B = 0.9 = 00223 = 22210 P5 2 5 1

75

52 + 0.1 !7

7 52 !7

490.0 =21 2

012223 2 4 1

2 : 0 0 2 2 301 01 !0 2

!0 1 1e

!2 1 1e11 !1

!3 1 21 !2 3

31

1e

) 22210 P = 0.9 ( 32200 +

A B 2: ' ' 2 . . -7 01 . X' 00:8 -20:8 - Y 30:8-10:8 , .X . 30:8-00:8 3 , ' 10:8-00:8?

':0141 ,01 = : !01 41 = e

41 = 7 2 =

': 4 = 2 2 = , : 4 = E X = V X ': ) P(A | B B,A :66

. ', 7.0? , 7.0 . ? : 1 , , , . , : lim (0.3 0.2) 0. 0. . x , ?X : x 1 . 8.0 3.0 + 7.0 = 1 = - P 1, , . : )1 = k-1 - = = 0.061 (, ( " ), . . :

A k 10:8-00:8 B 6 30:8-00:8 !6 6 = e

,6 =

6=32=

? ) = ( > - k = .0.06k

: z" , : )E Z = P(Z > 0=k

1: ' . 5 .

"

:

:

:

:

"

:

:

:

:

:

"

:

:

:

:

"

:

5,

3 5 2*1- . 6*7- 2 :

3 678901 = P

5

6721

: 3 , 3 -, . . 4 ? . 2 ? . ( ) 9 ,

9 = 3

8 2

6 2

4 2

2 2

!4

: . 549=

,549 = ) - P(A 1 - 3 2 9 2 3 2 !3 5 7 2 2 !4 3 2 3 2 3 2

-

.

, , 2 . 3 5 7 4 2 2 2 : 3 5 7 9 4 . . : , 1*2*3*4*5 : 1 04 , : , 5.0 5.0 . (2+ )I,i+1,i ( ) : Xi 1 2+ I,i+1,i . 83 1 = 1 = 5.0 5.0 52.0 5.0 83 =1

2

2

2

2

2 5.0 : __

) = (

= =

) 91 91 ( 5.0 91 + 52.0 91 =

61/3 = 57.0 52.0 4/1 = 52.0 52.0

1+ __ __ __ __ :J=i 2 2 , 2 , 2 =52.0. 5.0 52.0 52.0 5.0 52.0 = ) 1+Cov( ,

3 .1 11

1 8

2+ : J=I : I:1 8 11 44

=( Cov 2+ I,i. I :

2 , , 0 = = Cov , 2 22 , ..

0 81 + 61 81 + 8 73 2 + 4 91 + 61 91 =

3

1

1

1

1+J=i

2+ I=i 63 , "