-
..
, 2006
Copyright & A K-C
-
.
12 .
, , . .
, -
, . , .
.
.
.
1. .. . - .: , 1988.
2. .. . - .: , 1986.
3. .., .. . - .: ,
1982.
4. .. . - .: , 1987.
5. .., .. . - .: , 1988.
6. .., .., .. . - .: , 1980.
7. .., .. . - .: , 1986.
8. .., .. . .: , 1983.
9. .., .. . . - .:
- , 1962.
10. .., .. . - .: , 1969.
.
1 .
2 .
3 .
4 , .
5 .
6 . .
7 .
8 .
9 , .
10 , .
11 , .
12 , .
.
Copyright & A K-C
-
1
.
m
n , .. P(A)=mn . -
.
1.1. , 3 -1,0,1.
. , 1, -
?
:
1.
3 :
1) -1
2) 0
3) 1
n=3. (m=1)
P()= mn =
13
1.2. .
, .
:
n=18 ( ).
m=9
() = mn =
918 =
12
1.3. , 0 1. 0 1 -
.
0?
:
: 01, 10, 11, 00.
n=4,
m=3
P(A)=mn =
34 .
1.4. . : 6 -
, , , 3 ?
:
, , , , , . N=6. -
, 3 , 2 . , ()=16 , P(B)=
36 =
12 ,
P()=26 =
13
Copyright & A K-C
-
1.5. 2 , ,
5.
:
- 5.
n=36.
4
1 2 3 4
1- 1 2 3 4 , .. m = 4
2- 4 3 2 1
P(A)=mn =
436 =
19 .
1.6. 5- . : ) -
, ( : 15451); ) ; )
.
:
) n=99999-9999=90000 ( 1 99999 1
9999). 9 : 1, 2,,9,
10 : 0, 1, 2,,9, , , ..
m=9101011=900.
()=mn =
90090000 =
1100 .
) 9: 1, 2, 3,9, , 10: 0, 1, 2,,9;
: 5, 0. ( 5), .. m=91010102=18000, ()=mn =
1800090000 =
15 .
) 5 5 : 1, 3, 5, 7, 9
m=55555=3125
P(A)=mn =
312590000 =
5144 .
1.7. 10 4 . , -
?
:
10. , -
n=210
m=C24
()=mn = 2222
2222
10
4
C
=
215
1.8. 8 2, 4, 6, 8, 12, 7, 11, 13. 2
. , .
:
Copyright & A K-C
-
2 8. , -
28
8 : 2, 4, 6, 8, 12.
, 25
8=5+3
2=2+0 -
(A)=C25
C28 =
514
1.9. , 5 7
. 5 . -
: , : 1
, 2 2 ; ; D
.
:
5 15 (3+5+7).
, n=C515
15=7+8
A: 5=5+0 mA = C57
P(A)=mAn =
C57
C515 =
1143
15=3+5+7
B: 5=1+2+2 mB = C13 C
25 C
27
P(B)= mBn =
C31
52
72
155
C C C
=30143
15=3+12
C: 5=3+2 mC = C33 C
212 =1C
212 = C
212
P(C)=C212
C215 =
291
15=5+10
D:5=0+5 mD = C510
P(D)=mDn =
C510
C515 =
12143
1.10. ,,,, (6 49) 6 .
, : ) 6 ; ) 5 6; ) 3 ?
:
6 49. ,
649 =49!
6! 43! =13.983.816.
49=6+43
6 : 6=6+0
5 : 6=5+1
4 : 6=4+2
Copyright & A K-C
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3 : 6=3+3
(m1=1),
1+56 143 =1+
6!435! =259 (m2=259) ,
1+258+46C 243+C
36C
343=1 + 258 + 13545 + 246820 = 260624
:
(A)=m1 n =
1C 649
=1
13.983.816 0,00000007
(B)=m2 n =
1+C56 C 143
C 649 =
25913.983.816 =0,0000185
(C)=m3 n =
1+C56 C 143 +C
46 C
243 +C
36 C
343
C 649 =
260.62413.983.816 0,0186
1.11. , 3 , , ,
. , .
:
10 (0,1,
... ,9) 3.
n= 310 = 10!7! =720. m=1.
P(A)=mn =
1720 .
1.12. -
30 . . -
, , ?
:
30 5, -
n = 530 = 3029282726=17100720.
, , m=1.
(A)= mn =
1A530
= 1
17100720 .
1.13. , n , m .
. , -
.
:
m n ,
m , nm =mn.
, - m ( 1 , 2- ..
m-).
Copyright & A K-C
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P(A)= m
Anm =
mmn =
1mn-1 .
1.14. 10 , . -
, , 10 . -
, . :
10 10,
, 1010 =1010
, m=1010 =P10=10!
(A)= mn =
P10A1010
= 10!1010 0,00036
1.15. , 24 ?
( , 29 .)
:
365 24,
, A 24365 =36524.
, m=A 24365 A 24365
P(A)= A 24365 - A
24365
A 24365 =1 -
A 24365
A 24365 =1 -
365!341! 36524
1.16. 1,2,3,4,5 . ) -
. , ; ) -
3 , . , -
.
:
) 5 , .. n=P5
=5!
:
1) 2, 4 4 .
2) - 4, 4 4 , m=P4+P4=4!+4!
P(A)= P4+P4
P5 =
4!+4!5! =
24!5! =
25
) 5 3, .. n=A35 =543=60
:
1) - 2, 2 4 ,
2) - 4, 2 4 , ..
m=A24 +A24 =43+43=24
P(B)= A24 +A
24
A35 =
2460 =
25
1.17. 10 . , 3
.
Copyright & A K-C
-
:
n=P10=10!
, ,
, , 10, 8 ; 3
P3 , m=P3P8=3!8!
P(A) = mn =
P3P8P10
= 3! 8!10! =
115
1.18. 5 , , , , . , ,
,
:
3 (, , )
n = P(2, 2, 1) = (2+2+1)!2! 2! 1! =
5!22 = 30 m = 1
P(A) = 1
P(2,2,1) = 130
1.19. . ,
7 , , :
; ;
5; D 5, 1 2.
:
7 10 (0, 1, 2, 3, , 9). -
. n = A710 =107.
, 4 ( 10 ), 3
10 3. mA=310
10111=10310=104.
() = mAn =
A310 10 A710
= 104
107 = 0,001
, 7 , .. m=710 , ()=
mBn
= 710
710 =
10!3!107 0,0605.
, 5, 6
10 6.
mC=610 ,
()= mCn =
610 710
= 106
107 =0,1.
D , 5, 1 2,
3, 2, 2.
mD=(3, 2, 2)= (3+2+2)!3! 2! 2! =210
(D)= mDn =
210107 =2,110
-5=0,000021
Copyright & A K-C
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1.20. 2 . : 7
8 ?
: P1=636 >P2=
536
1.21. , , 14 : , , , , , -
, , , , ,
.
: 413
1.22. , 5 ?
: P = 37 , : 7 : 1 , 1 1
.
1.23. , 900 60 , -
, 180 ,
, 150 , 20
, . -
: - , - ,
, - , ?
: 115 ,
56 ,
29 .
1.24. , . ,
:
) , ) , .
: 190 ,
181 .
1.25. , 20 25 2 ,
25. , : ) ; ) ; )
.
: ) C220
C225 =
1930 , )
C120 C15
C225 =
13 , )
C25
C225 =
130 .
1.26. 40 , 10 . 2 .
, 5 ?
: C510 C
1530
C2040 =
139365040
1.27. 10 5 . 3 .
?
: , = C 210 C
15
C 315 0,494
1.28. 5 , 3 3 2 5 .
3 . , : ) ; ) 2
; ) 7 .
Copyright & A K-C
-
: ) 15 C
13 C
12
C310 =
14 , )
C 310 - C15 C
13 C
12
C310 =
34 , )
C15 C23 +C
25 C
12
C310 =
724
1.29. 10 2. , 5 -
: ) 1 ; ) ; ) .
: ) 12 C
48
C510 =
59 ) C510
38C =
29 )
C58
C510 =
29
1.30. ,
?
: 1
A46 =
1360
1.31. 6 .
. : = {
5 }; = { }; = { -
}.
: ) 6
A56 =
11296 )
A56 - 6A56
= 12951296 )
6P5A56
= 554
1.32. 10 , ,
8 , , -
. : 1 2 -
. . .
: ()= 8810
= 145 , ()=
38810
= 115 .
: 3 : 1 8, 2 9 3 10.
1.33. () -
, 4.
,
.
: 1
42 =
116 .
1.34. 6 , , , , , . ,
, ?
: 1
(3, 2,1) =
160 .
Copyright & A K-C
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2
()=()(/)=()(/), (/)
, .. ,
. ,
()=()()
(+)=()+()-()
, (+)=()+()
2.1. 15% . , 5
?
:
: i i- ( 5,1=i ); i - . -
5 .
()=(12345)=(1)(2)(3)(4)(5)=(,85)5,44.
2.2. , -
. , , -
0,3; 0,4; 0,6.
, ?
:
i-(i=1,2,3) , , i ; i - .
.
) ()=(123)=(1)(2)(3)=(1-(1)).
(1-(2))(1-(3))=(1-0,3)(1-0,4)(1-0,6)0,168.
) (1)=1.
() = (1 2 3) = (1) (2) (3) = 1 (1- (2)) (1- (3)) = 1 (1-0,4)
(1-0,6) =
0,24.
2.3. 96% ( ); -
75 ( ). ,
, .
:
( ), , , , ,
( ) ( ).
()=0,96, (/)=0,75.
()=()(/)=0,960,75=0,72.
2.4. , 20 25 2 , -
25. , .
:
:
i i- (i=1,2)
Copyright & A K-C
-
.
()=(12)=(1)(2/1)=2025
1924 = 30
19.
2.5. 6 , .
, . ,
.
:
:
1 P,
2 ,
3 - ,
4 - ,
5 - ,
6 - , i- , i =1 6, .
.
()=(123456)=(1)(2/1)(3/12).
(4/123)(5/1234)(6/12345)=16
25
14
13
12 1=
1360 .
2.6. 100 .
.
, 5% .
:
, ,
. . =12345 , ={ -
} =15, .
()=(1)(2/1)(3/12)(4/123)(5/1234)=95100
9499
9398
9297
9196 0,77
()=1-P(A)1-0,770,23.
2.7. 20 , 4 ( 5 -
). 4. : ={ -
}, ={ }.
:
:
1 ( 4 );
A2 , ;
3 , ;
4 , , .
()=(1234)=(1)(2/1)(3/12)(4/123)=
2020
1519
1018
517 0,13
i= i- , 1- (i=2,3,4)
Copyright & A K-C
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()=(1234)=(1)(2/1) (3/12) (4/123)=
2020
419
318
217 0,004.
2.8. , 5 10 , 5 3 . -
, .
:
:
i i- (i=1,5).
- 5 .
()=(12345)=(1)(2/1)(3/12)(4/123)
(5/1234)
15=5+10
1:3=1+2
1 n1=315
m1=C15 C
210
(1)=m1n1
=C15 C
210
C315
12=4+8
(A2/A1): 3=1+2 n2=C312 , m2=C
14 C
28
P(A2/A1)=m2n2
=C14 C
28
C312
9=3+6
(A3/A1A2): 3=1+2 P(A3/A1A2)=C13 C
27
C39
6=2+4
(A4/A1A2A3): 3=1+2 P(A4/A1A2A3) = C12 C
25
C63
3= 1+2
(A5/ A1A2A3A4): 3= 1+2 P(A5/ A1A2A3A4) = 1
P(A) = C15 C
210
C315
C14 C28
C312
C13 C26
C39
C12 C24
C36 10,08
2.9. .
,
0,01. , 0,01; - 1=0,08; 2=0,12; -
- 3=0,1. :
:
:
- , .. 0,01.
- , .. 0,01.
Copyright & A K-C
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i- i- 0,01 (i=1,2,3)
1- -, 2- - , 3- - .
()=1-()=1-(123)=1-(1)(2)(3)=1-(1-(1))(1-(2))(1-(3))=1-(1-0,08)(1-0,12)(1-
0,1)=1-0,920,880,90,27.
2.10. , ,
0,7 ?
:
:
;
;
i i- , i=1,2,3
()=1-()=1-(123)=1-(1)(2)(3)=1-(1-(1))(1-(2))(1-(3))=1-(1-,7)3=1-
,33=0,973.
2.11. .
, 3 . , , -
?
:
- 3 .
- 3 .
i - i- (i=1,2,3).
()=1-()=1-(123)=1-(1)(2/1)(3/12)=1-910
89
78 =0,3.
- 3 , , .
()=1-(1)(2/1)(3/12)=1-45
34
23 =0,6.
2.12. , 30 24. ,
?
:
- .
- 1-, 2- .
i - i- (i=1,2)
()=1-()=1-(12)=1-(1)(2/1)=1-630
529 =1-
129 =
2829 0,97.
2.13. . , -
, 0,95?
:
n .
n .
i i- i =1,n .
()=1-()=1-(12n)=1-(1)(2)(n)=1-(12 )
n.
Copyright & A K-C
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1-(12 )
n>0,95, (12 )
n20 n>log220 , n>lg20lg2 , n>
lg2+1lg2 (lg20,3)
n>4,3 .. n5.
2.14. , 100 , 5 , -
, -
50.
:
- .
- .
- .
- .
100=5+95 n= 50100
B:50=0+50C:50=1+49 m1=
5095 , m2=
4995
15 .
()=(+)=()+()=m1n +
m2n =
C5095
50100 +4995
15
50100 0,18.
2.15. 20 25 . , -
, 3 4 . , ?
:
- , .. 3 4 .
- 3 .
- 4 .
25=20+5 n =425 =12650, m1=C320
15 =5700
: 4 = 3+1 m2=C420 =4845
: 4 =4+0
()=(+)=()+()=m1n +
m2n =
C320 15 +
420
425 0,45+0,380,83.
2.16. , , 5 ,
11 , 8 , 10,8 6. .
, .
:
D
i i-
i i-
Di i- , i=1,2.
()=(++D)=()+()+(D)=(12)+(12)+(D1D2)=(1)(2)+(1)(2)+(D1)(D2)=
524
1024 +
1124
824 +
824
624 =
186576 0,32.
Copyright & A K-C
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2.17. -6
6 . : ) ; ) -
- .
- , .. 3 6
i - i- 3
i - i- 4
i - i- 5
Di - i- 6 , i=1,2
F - - .
Fi - i- - .
49=6+43 n=649
i: 6=3+ 3 m1=36
343
Bi: 6=4+2 m2 =46
243
Ci: 6=5+1 m3 =C56 C
143
Di: 6=6+0 m4= C66 =1
()=(12)=(1)(2)=m1n
m1n =
C
36 C
343
C649
2
0,0003115
(F)=P(F1F2)=P(F1)P(F2)=P(A1+B1+C1+D1)
P(A2+B2+C2+D2)=(P(A1)+P(B1)+P(C1)+P(D1))
(P(A2)+P(B2)+P(C2)+P(D2))=2)2( 431
n
m
n
m
n
m
n
m++++++++++++ =
C
36 C
343 +C
46 C
243 +C
56 C
143 +1
C649
2
0,0003473
2.18. 10 20 ., 5 15 . 2 10 . 6
. , ?
:
6 .
6 .
6 20 . (=1,20)
5 20 . 15 . (=1,15)
D 5 20 10 . (=1,10)
4 20 . 2 15 . (=1,10)
F 4 20 , 10 15 .(=1,05)
G 3 20 . 3 15 . (=1,05)
20 . 15 . 10 .
17 = 10 + 5 + 2
: 6 = 6 + 0 + 0
: 6 = 5 + 1 + 0
D: 6 = 5 + 0 + 1
E: 6 = 4 + 2 + 0
F: 6 = 4 + 1 + 1
G: 6 = 3 + 3 + 0
Copyright & A K-C
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()=1-()=1-(()+()+P(D)+P(E)+P(F)+P(G))=1-1
C617 (C610 +C
510 C
15 +C
510 C
12 +C
410 C
25 +C
410 C
15 C
12 +C
310 C
35 )0,4.
2.19. 2 , 3 .
: - - , - , - , D -
.
:
Ei i- .
P(A)=P(E1E2)=P(E1)P(E2)=26
36 =
16
P(B)=P(E1E2+E1E2)=P(E1)P( 2E )+P(E1)P(E2)=26 6
3+
46
36 =
12
P(C)=P(E1E2)=P(E1)P(E2)=46
36 =
13
P(D)=1-P(D)=1-P(E1E2)=1-46
36 =
23
2.20. . , -
, , 0,6; 0,7; 0,8. -
:
- 3-
-
C
D - 2-
- , 2-
:
i i- , i=1,2,3.
P(A)=P(A1A2A3)=P(A1)P(A2)P(A3)=0,60,70,8=0,336
P(B)=1-P(B)=1-P(A1A2A3)=1-P(A1)P(A2)P(A3)=1-0,40,30,2=0,976.
P(D)=P(A1A2A3+A1A2A3+A1A2A3)=P(A1)P(A2)P(A3)+P(A1)P(A2)P(A3)+P(A1)P(A2)P(A3)=0,60,70,2
+0,60,30,8+0,40,70,8=0,452.
P(C)=P(A1A2A3+A1A2A3+A1A2A3)=P(A1)P(A2)P(A3)+P(A1)
P(A2)P(A3)+P(A1)P(A2)P(A3)=0,40,30,8+0,40,70,2+0,60,30,2=0,188.
P(E)=P(D+A)=P(D)+P(A)=0,452+0,336=0,788.
2.21. 2 30
36 . . -
: - ; - , ; - .
:
Di i-
Ei i- , i=1,2.
P(A)=P(D1E1)=P(D1)P(E1/D1)=3066
2965 =
29143
P(B)=P(D1E2+D2E1)=P(D1)P(E2/D1)+P(D2)P(E1/D2)=3066
3665 +
3666
3065 =
72143
Copyright & A K-C
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P(C)=1-P(C)=1-P(A)=1-143
29=
143
114
2.22. . 3-
. 2. 5 -
, .
. , ?
:
, .. 2 3.
2
3
Di i-; i=1, 2, 3
P(A)=P(B+C)=P(B)+P(C)=P(D1D2D3+D1D2D3+D1D2D3)+P(D1D2D3)=
P(D1)P(D2)P(D3)+P(D1)P(D2)P(D3)+P(D1)P(D2)P(D3)+P(D1)P(D2)P(D3)=15
15
45 +
15
45
15 +
45
15
15 +
15
15
15
=34125 +
1125 =
13125 =0,104.
2.23. , -
, p=0,05525.
, : ) -
; ) .
:
, .
, -
.
Di i- , i=1, 2.
P(A)=P(D1+D2)=P(D1)+P(D2)=2p=0,1105
P(B)=P(D1+D2)=P(D1)+P(D2)-P(D1D2)=P(D1)+P(D2)-P(D1)P(D2)=p+p-pp=2p-p2=0,1075.
2.24. - 1
2, .
, .
- 1 2
1 2. , ?
:
i i-
i i- ; i=1,2
P(A)=P(BC)=P(B)P(C)=P(B1+B2)P(C1+C2)=((P(B1)+P(B2)-P(B1B2))-(P(C1)+P(C2)-P(C1C2))=(P(B1)+P(B2)-
P(B1)P(B2))(P(C1)+P(C2)-P(C1)P(C2))=(p1+p2-p1p2)2
1' 2' 1" 2"
Copyright & A K-C
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2.25. k1, k2, k3, k4,
q1, q2, q3, q4 (pi=1-qi, i=1 4, ). , ?
30
A) k2 k1 k3 k4
B) k1 k3 k2 k4
:
i i- , i=1 4,
) k2, k3, k4, .. k2,k3,k4.
P(A)=P(A1B)=P(A1)P(B)=P(A1)(1-P(B))=P(A1)(1-P(A2A3A4))=P(A1)
(1-P(A2)P(A3)P(A4))=p1(1-q2q3q4)
) 1 k1 k2, 2 k3 k4.
1)
P(A)=P(C1C2)=P(C1)P(C2)=P(A1+A2)P(A3+A4)=(P(A1)+P(A2)-P(A1A2))(P(A3)+P(A4)-
P(A3A4))=(P(A1)+P(A2)-P(A1)P(A2))(P(A3)+P(A4)-P(A3)P(A4))=(p1+p2-p1p2)(p3+p4-p3p4).
2)P(A)=P(C1)P(C2)=(1-P(C1))(1-P(C2))=(1-P(A1)P(A2))(1-P(A3)P(A4))=
(1-q1q2)(1-q3q4).
2.26. ) ) 2 ) - , - . -
, 0,7; -
.
A) 2 1 3 2' 1' 3'
B) 2' 1' 2 1 3 3'
:
.
i i , i=1 3,
Bi i i=1 3,
D 2 3
.
P(C)=P(A1D)=P(A1)P(D)=P(A1)(1-P(D))=P(A1)(1-P(A2A3))=P(A1)(1-P(A2)P(A3))=0,7(1-0,32)=0,673.
A ( ))
1 1, 2, 3.
Copyright & A K-C
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2 1, 2, 3
D2 2 3.
P(A)=1-P(A)=1-P(E1E2)=1-P(E1)P(E2)=1-(1-P(E1))(1-P(E2))=1-(1-P(A1D))(1-P(B1D2))=1-(1-
P(A1)P(D))(1-P(B1)P(D2))
P(D1)=1-P(D1)=1-P(A2A3)=1-P(A2)P(A3)=1-0,30,3=1-0,32
P(D2)=1-P(D2)=1-P(B2B3)=1-P(B2)P(B3)=1-0,32
P(A)=1- (1-0,7(1-0,32))2=0,868.
B ( )
F1 1, 1
F2 2, 2, 3, 3.
P(B)=P(F1F2)=P(F1)P(F2)=(1-P(F1))(1-P(F2))=(1-P(A1B1)
(1-P(A2A3B2B3))=(1-0,32)(1-0,34)=0,907.
P(A)P(C) =
0,8680,637 1,36,
P(B)P(C) =
0,9070,637 1,41.
%.
P(A)-P(C)P(C) 100=36%
P(B)-P(C)P(C) 100=43%
) .
2.27. N .
- , :
1 2 L1 L2 L3
0,6 0,5 0,4 0,7 0,9
L1 M K2 L2 K2 N L3 .
:
i (i=1,2) , i, - -
i, - Lj (j=1, 2, 3). -
=P(A+B)=P(A)+P(B)-P(A)P(B) ..
P(A)=P(A1)+P(A2)-P(A1)P(A2)=0,6+0,5-0,60,5=0,8.
P(B)=P(L1)(L2)(L3)=0,40,70,9=0,252,
p=0,8+0,252-0,80,252=0,85.
Copyright & A K-C
-
2.28. -
. ) , -
; ) - -
; ) - ; )
- .
: ) (1-)(1-) ) ) (1-) ) (1-)
2.29. ,
(1)=0,8, (0)=0,2. 3- .
: P(000)=0,23 P(001)=P(100)=P(010)=0,80,22
P(110)=P(101)=P(011)=0,20,82 P(111)=0,83
2.30. , 3 1, 2, 3, -
. -
12 (1-) -
. , .
, 111, 123.
: p12 (1-p)
12 (1-p).
2.31. , 1, 0, 0, 0, -
1000.
: 14 111=
14 .
2.32. ,
.
: 16
15
14
13 =
1360 .
2.33. 5 , , , , . ,
.
: 25
24
13
12 =
130
2.34. 6. ,
1 2 .
: 2531
2430 =
2031 .
2.35. 7 3 . 3 .
, .
: 710
69
58 =
724 .
2.36. , : 0 1. 0 1 -
. , 0?
: 1-12
12 =
34 .
Copyright & A K-C
-
2.37. 3 . 1-
0,15. 2- =0,1, 3- - 0,12. , ,
, .
: 1-0,850,90,88=0,37.
2.38. , 4 , 0,9 ( ).
, .
: 1-0,14=0,9999.
2.39. 0,9984. -
, .
: =0,8.
2.40. 0,5.
, 0,9.
: n4 (1-0,5n0,9).
2.41. , , 0,2. -
, 0,99 ,
.
: n3 (1-0,2n>0,99), lg 50,7.
2.42. , 12 -
. , : ) ; ) 6-, 2-; )
1- ; ) .
: ) 112
112 =
1144 , )2
112 12
1=
72
1
)1112
1112 =
121144 , )1-
121144 =
23144 .
2.43. 0,01; 0,006; 0,002 -
. , .
: 0,01+0,006+0,002=0,018.
2.44. . , , , 0,8.
, .
: 0,80,2+0,80,2=0,32.
2.45. , , , 2 :
. .
- 1 1 1=0,9 0 1-
1=0,1, 0 0 q1=0,8 1 1-q1=0,2. -
- p2=0,7; 1-p2=0,3; q2=0,6; 1-q2=0,4. -
:
10, ,
10, 2 .
:
P(A)=P(T11)P(T00)=[p1p2+(1-p1)(1-q2)][q1q2+(1-q1)(1-p2)]0,36.
P(B)=P(T11)P(T01)+P(T10)P(T00)=[p1p2+(1-p1)(1-q2)][q1(1-q2)+p2(1-q1)]+[p1(1-p2)+(1-p1)q2][q1q2+(1-q1)(1-
p2)]0,49.
Copyright & A K-C
-
2.46. 10 : , , -
, 2 - 5 - . : )
; ) ; )
: , , ; )
: , , .
: ) 0,3; 0,2;0,5
) 0,3+0,2=0,5.
) P(AB)=310
29 =
115 P(CA)=
510
39 =
16 P(CB)=
510
29 =
19
) P(ABC)=310
29
58 =
124 P(CAB)=
510
39
28 =
124 P(CBA)=
510
29
38 =
124
2.47. . ,
0,9, - 0,8. , ,
: ) ; ) .
: ) 0,90,90,8=0,648,
) 1-(0,90,10,2+0,10,90,2+0,10,10,8+0,10,10,2)=0,954.
2.48. . 2, 42, 44,
42, 44. . , : )
42; ) 42; ) .
: ) 13
12 =
16 , )
13 +
12 -
13
12 =
23 , )
13
12 +
13
12 =
13 .
2.49. . -
, , 0,5, - 0,9.
, , ?
: 0,5+0,9+0,90,5=0,95.
2.50. , . -
, . -
k k - ( qk=1-pk - ). -
, .
.
a) 1 b) 1 2 3 c) 2 1 4 2 4 5 6 3 3 : ) 1-q1q2q3
b) 1-(1-p1p2p3)(1-p4p5p6) ) p1p4(p2+p3-p2p3)=p1p4(1-q2q3).
Copyright & A K-C
-
3 . , -
kH () ),1( nk = . -
= =
==n
k
n
kkkk HPHAPHPAP
1 1
1)(),/()()(
)(),...(),( 21 HnPHPHP , - , "" , ..
)/( AHP K , ,)(
)/()()/(
AP
HAPHPAHP KKK
=
nk ,1= 3.1 ,
1f 2f , 1f 2f
. 1f 0,3; 1f -0,6. , ?
:
kH - ;kf =1,2.
kH .
2
1)()( 21 == HPHP ( 1f 2f ).
={ }; 6,0)/(;3,0)/( 21 == HAPHAP ;
=
=+==2
1
45,06,02
13,0
2
1)/()()(
kkk HAPHPAP
3.2 "1" () 5
1 "0"
( ) 5
4. ,
"1", "1" "0" 0,1; "0" "1" -0,3. :
: 1H - "1"
2H - "0" = { "1"}
5
1)( 1 =HP 9,0)/( 1 =HAP ("1" "0")
5
4)( 2 =HP 3,0)/( 2 =HAP ("0" "1")
=
==+==2
1
42,050
213,0
5
49,0
5
1)/()()(
kkk HAPHPAP
3.3 , . , - 2:3:5. , 0,05; 0,1; 0,2. . , .
:
: kH - - , -1, 2, 3. = { } .. 2:3:5
10
2)( 1 =HP
Copyright & A K-C
-
95,005,01)/( 1 ==HAP
10
3)( 2 =HP 9,01,01)/( 2 ==HAP
10
5)( 3 =HP 8,02,01)/( 3 ==HAP
=
=++==3
1
86,08,05,09,03,095,02,0)/()()(k
kk HAPHPAP
3.4 12 10 , . , , , -. , , .
: : 1- ; 2- ;
12
1)( 1 =HP 12
11)( 2 =HP
-
11
2)/( 1 =HAP ( 10 , 1 , 1 )
11
1)/( 2 =HAP ( 10 , 1 , )
=
=+==2
1 132
13
11
1
12
11
11
2
12
1)/()()(
kKK HAPHPAP
3.4 15 , . -
25 . , , - 2 1 ?
:
KH (=0,1,2), .
KH .
k
k n
m =)( 230Cnk =
( 30 25) 30=25+5
0H : 2=0+2 250 Cm =
1H : 2=1+1 15
1251 CCm =
2H : 2=2+0 2252 Cm =
29
20
!30!23!2
!28!2!25)(
87
25
!30
!28!2525525)(
87
2
!30!3!2
!28!2!5)(
230
225
2
230
230
15
125
1
230
25
0
===
===
=
===
C
CHP
CC
CCHP
C
CHP
= { } 0)/( 0 =HAP 1)/( 2 =HAP
7
6
28
24)/( 1 ==HAP ( 28 , 24 ).
53,0609
3251
87
25
7
6
87
250
87
2)/()()(
2
0
=++== =k
kk HAPHPAP
Copyright & A K-C
-
3.6 0,8 , 0,2 - . , - 0,7; - 0,3. , - -. , .
:
: 1- , 2- . = { - }.
8,0)( 1 =HP 7,0)/( 1 =HAP 2,0)( 2 =HP 3,0)/( 2 =HAP
:
903,031
28
3,02,07,08,0
7,08,0
)/()(
)/()()/(
2
1
111 =+
=
=
=kkk HAPHP
HAPHPAHP
3.7 60% - , 80% 75% . - - . , .
: : 1- -
2 -- (1)=0,6 (60% 100% -) (2)=0,4 = { } P(A/H1)=0,8 (80% )
P(A/H2)=0,75 (75% ) :
13
8
75,04,08,06,0
8,06,0
)/()(
)/()()/(
2
1
111 =+
=
=
=k
kk HAPHP
HAPHPAHP =0.62
3.8 "" "."
, 5
2"" "".
, "" "" 5:3. - , , : ) ""; ) "".
: - "", - "". : H1- "", H2- "". P(H1) : P(H2)=5;3, ,
P(H1)+P(H2)=1 P(H1)=5/8; P(H2).=3/8 P(A/H1)=3/5; P(A/H2)=1/3;
P(B/H1)=2/5; P(B/H2)=2/3.
====
====++++========2
1 2
1
3
1
8
3
5
3
8
5)/()()(
lii HAPHPAP
====
====++++========2
1 2
1
3
2
8
3
5
2
8
5)/()()(
lii HBPHPBP
4
3
2
15
3
8
3
)(
)/()()/( 111 =
==
AP
HAPHPAHP
Copyright & A K-C
-
2
1
2
13
2
8
3
)(
)2/()2()/( 2 =
=
=
BP
HBPHPBHP
3.9 25 , , 10 , 7 , 5- 3 . 25 , - 20, -15 10 . . :
1S = { }; 2S = { }; 3S = {-
}. :
: H1- H2 - H3 - H4 - . P(H1)=10/25, P(H2)=7/25,P(H3)=5/25,
P(H4)=3/25 = { }. P(A/H1)=1, B k - - , -1,2.
30
19
24
19
25
20)/()()()/( 121212 ==== BBPBPBBPHAP
20
7
24
14
25
15)/()()()/( 121213 ==== BBPBPBBPHAP
20
3
24
9
25
10)()/( 214 === BBPHAP
:
6,0499
300
20
3
25
3
20
7
25
5
30
19
25
71
25
10
125
10
)/()(
)/()()/(
4
1
111
=
+++
=
=
=kkk HAPHP
HAPHPAHP
03,0998
27
750
49920
3
25
3
)(
)/()()/()(
11,0998
105
750
49920
7
25
5
)(
)/()()/()(
87,0)/()/()(
27,0499
133
750
49930
19
25
7
)(
)/()()/(
)750
499)/()())((
4443
3332
211
222
4
1
=
=
==
=
=
==
+=
=
=
=
== =
AP
HAPHPAHPSP
AP
HAPHPAHPSP
AHPAHPSP
AP
HAPHPAHP
HAPHPAPk
kk
3.10 . , 1- 2- , , - .3,0;4,0;2,0 321 === ppp
: : 1H -3-
Copyright & A K-C
-
2H -3- .
7,0)(;3,0)( 21 == HPHP = { }, = {- } =1,2
.08,04,02,0)()()()/(
44,06,02,04,08,0
)()()()()()/(
21212
212121211
====
=+=+=+=
BPBPBBPHAP
BPBPBPBPBBBBPHAP
47
14
188,0
056,0
08,07,044,03,0
08,07,0
)/()(
)/()()/(
2
1
222 ==+
=
=
=k
kk HAPHP
HAPHPAHP
3.11 , , , 10 , - - . ) , ? ) , - , .
:
: 1H -
2H 3H
10
2)(,
10
5)(,
10
3)( 321 === HPHPHP
= { }
3
2)/( 1 =HCP ( 3 , 2 )
2
1)/( 2 =HCP 3
1)/( 3 =HCP
) =
=++==3
1 60
31
3
1
10
2
2
1
10
5
3
2
10
3)/()()(
kkk HCPHPCP
)31
15
60
312
1
10
5
)(
)/()()/( 2
22 =
=
=
CP
HCPHPCHP
3.12 . - 5 3 , 4 4 . - . ) , ? ) , ?
:
: kH - , =1,2,3 3
1)( =kHP
={ }
2
1
8
4)/(
8
5)/()/(
3
21
==
==
HAP
HAPHAP
12
7
2
1
3
1
8
5
3
1
8
5
3
1)/()()(
3
1
=++== =k
kk HAPHPAP
Copyright & A K-C
-
14
5
12
78
5
3
1
)(
)/()()/()/( 1121 =
=
==
AP
HAPHPAHPAHP
-
7
5
14
5
14
5)/()/()( 21 =+=+= AHPAHPBP
3.13 : , , . , 0,3:0,4:0,3. 0,6. , , 0,2. , . - , , .
: : 1H -
2H -
3H -
.3,0)(;4,0)(;3,0)( 321 === HPHPHP
D={ }, -- 4,1=k
.0048,02,06,02,02,0)()/(
0048,02,02,06,02,0)()/(
0144,06,02,02,06,0
)()()()()()/(
43213
43212
432143211
===
===
=====
EEEEPHDP
EEEEPHDP
EPEPEPEPEEEEPHDP
16
9
0048,03,00048,04,00144,03,0
0144,03,0
)/()(
)/()()/(
3
1
111
=++
=
=
=
=k
kk HDPHP
HDPHPDHP
3.14 ,
25,0;5,0;25,0 321 === . , , 0,1;0,2 0,4 . , . : 0,225. 3.15 3 .
20% , -10% - 5%. , - 30 % , 20% - 50% . : 0, 895. 3.16 , -, . 0,06,
- 0,03. ? : 0,05. 3.17 5 4 . , . , ( ). : 5/9. 3.18 24 5 . , -,
0,9. 0,65. - . , ? : 90/337.
Copyright & A K-C
-
3.19 (), () () . - , 40% . 60% . , 0,02. - 0,03 0,04. , , -. , ?
: 8/29. 3.20 , . , 1- , 1-, 2- 3- - 0,4; 0,3; 0,5. : 20/29. 3.21 3
4/5, 3/4, 2/3 . 3 - 2 ( ). - , . : 6/13. 3.22 , , , 1, - 2. 1 3 1 .
2 1 3 . ) , ? ) , 1- , ? : ) 1/2; ) 3/4.
3.23 , (), (), (), 3:2:5. , 0,8; 0,9; 0,9. ) , ? ) , . : ) 0,87;
) 18/87. 3.24 , -. - 0,8, -0,7, - 0,5. ) - . ) . , . : ) 0,47; )
12/47.
Copyright & A K-C
-
4 , .
,
, .
()
.
.
ix - ip .
1X 2X nX
1p 2p np
11
==
n
iip
( ) F(x) , -
P (X
-
m x px kk
n
k= =
1
, -
( )m x f x dxx =
+
.
, , , .
X :
( )[ ] [ ] [ ]( )D M X m M X M Xx x= = 2 2 2
( ) ( )D x m p x p mx k xk
n
k k k xk
n
= = = =
1
22
1
2
( ) ( ) ( ) ( )x x xD x m f x dx x f x dx m= =
+
+
2 2 2
C X x xD= .
0 . -
0 ,
.
,
:
( ) ( )P X M P X Me e< = > . 4.1 X
, , .
:
C X 0, 1 2 .
Bk ={ k- } k=1 ,2 .
( ) ( ) ( ) ( )P X P B B P B P B= = = = =0 0 5 0 5 0 251 2 1 2 ,
, , ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
P X P B B B B P B P B P B P B
P X P B B P B P B
= = + = + = + =
= = = = =
10 5 0 5 0 5 0 5 0 5
2 0 5 0 5 0 25
1 2 1 2 1 2 1 2
1 2 1 2
, , , , ,
, , ,
X 0 1 2 P 0,25 0,5 0,25
( )F x
x
x
x
x
=
< < >
0 00 25 0 10 75 1 21 2
,
, ,
, ,
,
4.2 . ,
, 0,9. ,
Copyright & A K-C
-
. : ) -
X , ; ) -
K0 .
:
Bk ={ k } k=1,2,3, ...
( ) ( )( ) ( ) ( ) ( )( ) ( )
( ) ( )
P X P B
P X P B B P B P B
P X P B B B
P X k P B B B Bk kk
= = = =
= = = = =
= = = =
= = =
1 1 0 9 0 1
2 0 9 0 1 0 09
3 0 9 0 1 0 081
0 9 0 1
1
1 2 1 2
1 2 32
1 2 11
, ,
, , ,
, , ,
.................................................................
... , ,
X 1 2 3 ... k .... k - 1 P 0,1 0,09 0,081 .... 0,9 - 0,1 k0 = 1
( ) .
4.3 4 . 0,5 -
, . -
X - , .
:
Bk ={k- } k=1,2,3,4.
X 0 1 2 3 4 P 0,5 0,25 0,125 0,0625 0,0625
4.4 3 .
0,1 . -
X , ,
, . -
, X , ( )F x , - : m Dx x x, , .
:
X , . X :
X 0 1 2 3 P 0,729 0,243 0,027 0,001
Copyright & A K-C
-
( ) ( ) ( ) ( ) ( )f x x x x x= + + + 0 729 0 243 1 0 027 2 0
001 3, , , , .
( ) ( )F x P X x
x
x
x
x
x
= < =
< < <
>
0 0
0 279 0 1
0 972 1 2
0 999 2 3
1 3
,
, ,
, ,
, ,
,
4.5 7 , 4 , . -
3 . X .
X { X 2 } :
( )( )( )( )
7 4 3
0 3 0 3
1 3 1 2
2 3 2 1
3 3 3 0
= += = +
= = +
= = +
= = +
X
X
X
X
:
:
:
:
( )P X k mn
n C k
m C
m C C C
m C C C
m C
k= = = =
= =
= =
= =
=
, , ,73
0 33
1 41
32
32
2 42
31
42
3 43
0 3
1
4
3
( )
( )
( )
P P XC
P P XC
C
P P XC
C
0
73
132
73
242
73
01 3 4
7
1
35
14 4 3 3 4
2 7
12
35
23 3 4 3 4
2 2 7
18
35
= = = = =
= = =
=
=
= = =
=
=
! !
!
! ! !
! !
! ! !
! ! !.
Copyright & A K-C
-
4.6 10 2 . 3 . X 3 . - F(x) f(x) , mx Dx . :
( )( )( )
10 2 8
0 3 0 3
1 3 1 2
2 3 2 1
= += = +
= = +
= = +
X
X
X
:
:
:
( )
n C
P X km
n
m C m C C C
m C C
k
=
= =
= = =
= =
103
0 83
1 21
82
82
2 22
81
2
8
,
.
X 0 1 2
P C
C83
103
7
15=
C C
C82
21
103
7
15
=
C C
C81
22
103
1
15
=
( )F x
x
x
x
x
=
<
<
>
0 0
7
150 1
14
151 2
1 2
,
,
,
,
( ) ( ) ( ) ( )f x x x x= + + 715
7
151
1
152
m x px k kk
= = + + ==
0
2
07
151
7
152
1
15
3
5
4.7 5 , .
X, ,
. .
:
Bk = { k }, k = 1 5,
X 1 2 3 4 5 P 0,2 0,2 0,2 0,2 0,2
Copyright & A K-C
-
( )F x
x
x
x
x
x
x
=
< < < <
>
0 1
0 2 1 2
0 4 2 3
0 6 3 4
0 8 4 5
1 5
,
, ,
, ,
, ,
, ,
,
( ) ( ) ( ) ( ) ( ) ( )( )f x x x x x x= + + + + 0 2 1 2 3 4 5,
. 4.8 4 1 4 .
63
2 . X .
X , ( )F x , f(x) -: m Dx x x, , .
:
1 2 3 4 5 6
1 1 1 2 2 3
2 3 4 3 4 4
3 4 5 5 6 7
( ) ( ) ( ) ( )
( )
P Xm
nP X P X P X
P X
= = = = = = = = = =
= =
31
64
1
65
2
6
1
36
1
6
71
6
, , ,
.
X 3 4 5 6 7
P
( ) ( ) ( ) ( ) ( ) ( )
( )
f x x x x x x
m x p
D x p m
D
x k kk
x k k xk
x x
= + + + +
= = + + + + = =
= = + + + + =
= =
=
=
1
63
1
64
1
35
1
66
1
67
31
64
1
65
1
36
1
67
1
6
30
65
91
616
1
625
1
336
1
649
1
625
5
3
5
3
1
5
2 2
1
5
16
16
13
16
16
Copyright & A K-C
-
4.9 X 3 x1 =4 p1
=0,5 ; x2 =6 p2 =0,3 x3 c p3. x3 p3 mx =8.
:
.. pkk= =
1
3
1 ,
m x p x
x
x k kk
= = + +
==
1
3
3
3
8 4 0 5 6 0 3 0 2
21
, , , ,
.
4.10 ( ) 1,2,3,4 5 0,07 ; 0,21 ; 0,55 ; 0,16 ; 0,01. - 30 . C
?
: X X 1 2 3 4 5 P 0,07 0,21 0,55 0,16 0,01
( )
m x p
D x p
x k kk
x k kk
= = + + + + =
= = + + + +
+ = =
=
=
1
5
2
1
52 2 2 2
2 2
1 0 07 2 0 21 3 0 55 4 0 16 5 0 01 2 83
1 0 07 2 0 21 3 0 55 4 0 16
5 0 01 2 83 8 67 8 0089 0 6611
, , , , , ,
, , , ,
, , , , ,
( ) ( )
x x
x x
D
n m
= =
= + = + =
0 82
30 2 83 0 82 30 109 5 110
,
, , ,
. 4.11 C 3 I , II , III .
P I P II P III
1 0,5 0,3 0,2
2 0,4 0,4 0,2
3 0,4 0,5 0,1
, I 5 , II 3 , III 1.
:
C Xk k k = 1,2,3 .
X1 1 3 5 P 0,2 0,3 0,5
X2 1 3 5
P 0,2 0,4 0,4
X3 1 3 5
P 0,1 0,5 0,4
1 3 , .. [ ] [ ] X X3 1< . 2 1 ,
[ ][ ][ ]
M X
D X
X
1
12
1
3 6
15 4 3 6 2 44
1 56
=
= =
=
,
, , ,
,
[ ][ ][ ]
M X
D X
X
2
2
2
3 4
2
1 41
=
=
=
,
,
[ ][ ][ ]
M X
D X
X
3
32
3
3 6
14 6 3 6 1 64
1 28
=
= =
=
,
, , ,
,
Copyright & A K-C
-
3 2 , .. [ ] [ ] [ ]M X M X M X2 1 33 4= < =, 4.12 F1(x)
F2(x) X1 X2 c, a1
a2 , 1 . ,
( ) ( ) ( )F x a F x a F x= +1 1 2 2 X . :
1) F1(x) F2(x) , a1 a2 0 ;
a1F1(x) a2 F2(x) , .
) ( ) ( ) ( )( ) ( ) ( )
2 0 0 0
1 1 1
1 1 2 2 1 2
1 1 2 2 1 2
lim lim lim
lim lim limx x x
x x x
F x a F x a F x a a
F x a F x a F x a a
+ + +
= + = + =
= + = + =3) -
x ( ) ( ) ( ) ( ) ( ) ( )F x a F x a F x a F x a F x F x = + = +
=0 0 01 1 2 2 1 1 2 2 .
3 , .. ( )F x .
4.13 X .
( )F x A B arctgxa
x= + < < , ( ) , A B const .
a) , ( )f x . :
( )lim lim ,x x
F x A Barctgx
aA
BA
B
= +
= =
2 2
0
( )lim lim ,x x
F x A Barctgx
aA
BA
B
AB
A
AB
B
+ += +
= + + =
= =
+ = =
2 21
20
1
2
21
1
( )F x arctg xa
= +12
1
.
( ) ( ) ( )f x F x xa
a
a
a x= =
+ =
+1 1
1
12
2
2 2 .
4.14 X
( )F x
x
x x
x
=
<
>
06
36 3
13
,
cos ,
,
( )f x ,
(((( )))) .4
,4
,0,4
0
-
: (((( )))) (((( ))))
>>>>
-
2
3)2(1
2
1
2
3)2(
2
112
3)2(102
3
0 1
0 1
2)2(2
322
30)(,21)3
0
0 2
32
2
30)(,10)2
00)(,0)1
.)()(
2
3,1
3
2
0 1
0
2
1 2 3
202)2(20)(
xxxtt
xdttdttdtxFx
x xdttdtxFx
xdtxFx
xdttfxF
aa
x adxdxxadxaxdxdxxf
=+==
=
++=>>>
++++====++++++++====>
-
++
++=
=+=+=
===+=
==
>>>
>>>>
>>>>
-
7.31. (X,Y),
f x yx y
( , )( )
=+ +
2
12 2 3 :
0== yx mm
=
5,00
05,0k
Copyright & A K-C
-
8. .
, (X,Y) , -
+
=
2
2
2
2
2
2
)())((2)(
)1(2
1exp
12
1),(
y
y
yx
yxxy
x
x
xy
xyyx
mymymxrmx
r
ryxf
),( yx mm - , yx , - X,Y, xyr - . : (X,Y) -
( xyr =0), . X Y f(x,y)
f x yt
E Et
x m
E
y m
Ex y
x
x
y
y
( , ) exp( ) ( )
=
+
22
2
2
2
2,
2xx tE = , 2yy tE = (t0,4769...) - X Y . , , , :
=
y
y
y
y
x
x
x
x
mm
mmYXP
),(
,
()- , , :
137
( ) ( )( ) ( )x m r x m y m y mkx
x
xy x y
x y
y
y
+
=
2
2
2
22
2
xyr =0 (X,Y) D
yx kbka == , 22
1)),((k
eDYXP
=
== yx , . 0== yx mm R (X,Y) ( ) :
f rr
e r
r
r
( ) ,
,
= >
>>>
-
9.15. X
X -1 0 1 2 P 0,2 0,1 0,3 0,4
Y X= 2 . :
m P X x
D P X x m
yx
kk
yx
kk y
k
k
= = = + + + =
= = = + +
+ + =
=
=
2 2 0 2 2 01 2 0 3 2 0 4 2 4
2 2 0 2 2 01
2 0 3 2 0 4 2 4 1 99
1 0 1 2
1
4
2
1
42 1 2 0 2
1 2 2 2 2
( ) , , . , ,
( ) ( ) ( ) ( ) , ( ) ,
( ) , ( ) , ( , ) ,
9.16. X - 6 . - Y X= +2 1.
: .. X - 6 3 . :
P X k C p q n pnk k n k( ) , , , .= = = = q =
56
316
X 0 1 2 3 P 1 2 5
2 1 6
75
216
15
216
1
216
P X c p q
P X c p q
P X c p q P X c p q
m x P X xy k k
( )
( ) .
( ) , ( )
.
( ) ( ) ( )
= = =
=
= = =
=
= = =
= = = =
=
=
= + = = +
05
6
125
216
1 31
6
5
6
75
216
2 31
6
5
6
15
2163
1
6
1
216
2 1 2 0 1
30 0 3
3
31 1 2
2
32 2 1
2
33 3 0
3
+ + +
+ + + + =
= + = = + +
+ + + + + + =
=
=
125
2162 1 1
75
216
2 2 115
2162 3 1
1
2162
2 1 2 0 1125
216
2 1 175
2162 2 1
15
2162 3 1
1
2164
5
108
1
4
2 2 2
1
4
2 2 2
( )
( ) ( ) .
( ) ( ) ( ) ( )
( ) ( ) ( ) .
k
y k k yk
D x P X x m
9.17. X
f xx x
x( )
exp( ), ( )
,=
> >
0 0
0 0.
eY ==== .
Copyright & A K-C
-
:
22
2
2
2)2(
0
2222
)1)(2()1(2)1(0
2
1)()()(
++++++++====
++++
++++====
++++
++++
++++====
====
++++========
++++
++++
++++
x
xxy
xy
e
dxeemdxxfeD
9.18. r 2 , . . :
.. , , - (r;0). , , OX --, .
0 2 ,
f ( )
, ( , )
, ( , )
=
0 0 2
1
20 2
Sr
=2
2sin ;
M S S f dr
dr
d[ ] ( ) ( ) sin sin= = = =
+
2
0
2 2
02
1
2 42
= =r r2 2
2 0
cos
9.19. , R .
: .. , , - (R,0). , , -
OX.
2
2
,
f ( ),
,
==
+
1 1
0
22 2 2
2 2 2
Y 1 4 P 0,6 0,4
X 1 3 P 0,3 0,7
Copyright & A K-C
-
m x y f x y dx dy x yR
dx dy
Rd d
R
Rd R
D x y f x y dx dy m x yR
dx dy
RR
d d
ZD
R
Z ZD
R
= = + =
= = =
= = +
=
+
+
( , ) ( , )
.
( , ) ( , ) ( ) ( )
2 22
22
2
3
0
2
00
2
2 2 2 22
22
3
0
1
1 1
3
2
3
1
4
9
1
4
9 2
4
9 182
2 2 2
0
2
RR R R
= = .
9.23. 2 X Y. X
f x ex
1
2
181
3 2
2
( ) .( )
=
Y (1;5). :
M X Y M X Y M X Y M X M Y
M X Y M X Y D X Y D X Y
D X Y D X Y
[ ], [ ], [ ], [ ], [ ],
[ ], [ ], [ ], [ ],
[ ], [ ].
+ + + +
2 3 4
2 3 3 2 1
2 6
2 2
2 2
: m Dx x= =2 9, . Y:
m Dy y=+
= =
=1 5
23
5 1
12
4
3
2
,( )
.
,
.3
295923
3
4223
4922][ ====++++++++====++++++++==== xDymyDxmyDxDYXD
9.24. X Y, Y X= +5 3. - X : m Dx x= =2 1, .
: ) Y; ) X Y. :
Copyright & A K-C
-
a M Y M X m
D Y D X D X
x) [ ] [ ]
[ ] [ ] [ ]
= + = + = + =
= + = = =
3 5 3 5 3 5 2 13
3 5 5 25 1 252
k M X Y M X M Y M X X
m M X D X M X
xy
x
) [ ] [ ] [ ] [ ]
[ ] ( [ ] [ ])
( ) .
= = + =
= + = + + == + =
3 5 2 13
3 5 26 6 5 26
5 1 4 20 5
2
2 2
9.25. X f x ex
( ) =1
2
2
22
.
Y X X= +1 3 42 3 . :
m Dx x= =02, .
M X x f x dx X X[ ] ( ) ( ) , ( )3 3= =
+
M X x f x dx x e dxx
[ ] ( )3 3 3 21
20
2
2= = =
+
+
( )
M Y M X X M M X M X
D X M X M X
[ ] [ ] [ ] [ ] [ ]
( [ ] [ ]) [ ] ( )
.
= + = + =
= + + = + + =
=
1 3 4 1 3 4
1 3 4 1 3 0 4 0
1 3
2 3 2 3
2 3 2
2
9.26. X Y:
X Y . M X Y[ ],3 2 4+ M X Y[ ], D X Y[ ].3 2 5 +
: -
f x
xx
xf y
y y
y1 28
0 4
0 0 4
1
18
1
30 6
0 0 6( )
, [ ; ]
, [ ; ], ( )
, [ ; ]
, [ ; ]=
=
+
.. X Y:
m x f x dx xx
dxx
D x f x dx m xx
dx
x
x x
= = = =
= = =
+
+
1
3
0
4
21
2 2
0
4
8 24
4
0
8
3
8
64
9
( )
( ) ( )
= =x4
32
4
0
64
9
8
9
f x1 ( )
1
2
x 4
f y2 ( )
13
y 6
Copyright & A K-C
-
m y f y dy y y dyy y
D y f y dy m y y dy
y y
y
y y
= = +
= +
=
=
= = +
=
= +
=
+
+
2
3 2
0
6
22
2 2
0
6
4 3
1
18
1
3 54 6
6
0
2
1
18
1
34
72 9
6
0
4 2
( )
( ) ( )
M X Y m mx y[ ] .3 2 4 3 2 4 38
32 2 4 8+ = + = + =
M X Y m mX Y
kx y xy[ ]
( = = = =
8
32
16
3 0)
, -
D X Y D Dx y[ ] ( ) .3 2 5 3 2 98
94 2 162 2 + = + = + =
172 9.27. , 2 3 , . -
Y X X= +2 12 , X- , . :
.. X. X 0 1 2 P 0,3 0,6 0,1
P XC
CP X
C C
CP X
C
C( ) , ; ( ) , ; ( ) ,= = = = =
= = = =0 0 3 1 0 6 2 013
2
52
21
31
52
22
52
m x p
D x p m
M Y M X X M X M X D M X
x k kk
x k k xk
X
= = + + =
= = + + =
= + = + = +
+ = + + =
=
=
0 0 3 1 0 6 2 01 0 8
0 0 3 1 0 6 2 01 0 8 0 36
2 1 2 1 2
0 8 1 2 0 36 0 8 0 8 1 2
1
3
2 2
1
32 2 2 2
2 2 2
2
, , , ,
( ) , , , , ,
[ ] [ ] [ ] [ ] ( [ ])
, ( , , ) , ,2
9.28. 3 . , 3
n . Y X X= + 2 3 22 , X - .
:
A, P A C p q( ) = 32 2 1, p =
1
2, q =
1
2.
P A p( ) =
= =3
1
2
1
2
3
8
2
.. X- A n , A
p . .. X.
Copyright & A K-C
-
= + 27
64
93
6422n n .
9.29. (X,Y) :
m m rx y x y xy= = = = =1 1 2 3 0 5, , , , , . Z X Y= ( )2 .
Z.
:
M Z M X Y XY M X M Y M XY
D X M X D Y M Y M X M Y
rxy x y
[ ] [ ] [ ] [ ] [ ]
( [ ] ( ]) ( [ ] [ ]) ( [ ] [ ]
)
= + = + =
= + + + ++ =
2 2 2 2
2 2
2 2
2
11
, , k rxy xy x y= . 9.30. X Y [ ; ]01 ,
. - .
:
R Y X= ,
{ }M R M Y X[ ] .= ( )X Y, XOY, - f x y( , ) = 1 1.
y 1
D 2
D 1
x 0 1 174
D X Y Y X X Y1 : > = ; D X Y X Y X2 : Y > = .
M R x y dxdy y x dxdy
dx x y dy dy y x dx
DD
x y
[ ] ( ) ( )
( ) ( ) .
= + =
= + =
1 1
1
3
21
0
1
00
1
0
M R M Y X M Y XY X
M Y M X M Y M X D Y
M Y M X M Y D X M X
[ ] [ ] [ ]
[ ] [ ] [ ] [ ] ( [ ]
[ ]) [ ] [ ] ( [ ] [ ])
2 2 2 2
2 2
2 2
2
2
2
1
12
1
42
1
2
1
2
1
12
1
4
1
6
= = + =
+ = +
+ + + =
= +
+ +
=
R x 0 X Y 1
Copyright & A K-C
-
M X M Ya b
D X D Yb a
[ ] [ ] , [ ] [ ]( )
( )
= =+
=+
= = =
=
=
=
2
0 1
2
1
2 121 0
121
12
2
2
9.31. (X,Y) ( ; )11 3 2
2 4
. Z X Y= +2 4 3.
: Dx = 3 , D y = 4 , kxy = 2 , mx = 1, my = 1.
D Z D X Y D X Y D X D Y
kxy
[ ] [ ] [ ] [ ] ( ) [ ]
( ) ( ) .
= + = = + ++ = + =
2 4 3 2 4 2 4
2 2 4 4 3 16 4 16 2 108
2 2
9.32. 2 , 4. - 3. 3 . - .
: m mx y= = 0 , D Dx y= = 4, kxy = 3. Z X1 3= , Z Y2 3= . k M Z
Z M Z M Z M X Y M X M Y
M X Y M X M Y M X Y M X M Y k
Z Z
xy
1 2 1 2 1 29 3 3
9 9 9 9
9 3 27
= = =
= = = == =
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] ( [ ] [ ] [ ])
.D Z D X D X D Z[ ] [ ] [ ] [ ]1
223 3 9 4 36= = = = =
rk
Z Z
Z Z
Z Z1 2
1 2
1 2
27
36 36
27
36
3
4=
=
= =
.
9.33. [1,2] (), -, [10,12] (). , - .
: .. X - , X:
ma b
x =+
=+
=2
1 2
2
3
2.
Db a
x =
=
=( ) ( )2 2
12
2 1
12
1
12.
.. Y- , .. Y: my =+
=10 12
211.
.. Z- Z X Y= 2 m M X Y M X M YZ = = =[ ] [ ] [ ]
2 2
X Y
= + = +
=( [ ] [ ]) [ ]D X M X M Y2
1
12
9
411
77
3.
9.34. X Y: f xx
x10 5 0 2
0 0 2( )
, , [ ; ]
, [ ; ]=
;
f ye y
y
y
2
0
0 0( )
,
,=
>
======== xe
xtxFtm
xx
10.16. ( )X t
f x x t t ex t x t
2 1 2 1 22
1
2
1 12
2 22
( , , , )( sin ) ( sin )
=
+ + +
. , -
.
: m t t D t k t tt t
t tx x x( ) sin , ( ) , ( , )
,
,= = =
=
1
0
11 21 2
1 2
.
10.17. f x x t t ex t x t
2 1 2 1 22
12
1 12
2 22
( , , , )( ) ( )
=
. , .
: m t t D t k t tt t
t tx x x( ) , ( ) , ( , )
,
,= = =
=
1
0
11 21 2
1 2
.
V 0 1 2 P 0,5 0,2 0,3
Copyright & A K-C
-
11 , .
X(t), Y(t) Z(t) - , (t) , : 1. Z(t)=X(t)+Y(t) ,
M[Z(t)]=M[X(t)]+M[Y(t)]
kz(t1,t2) = kxy(t1,t2) + kyx(t1,t2) + ky(t1,t2) + kx(t1,t2),
kxy(t1,t2) -
X(t) Y(t). ( ) ( ) ( )
= 2121 tYtXMt,tk xy
, kxy(t1,t2) = kyx(t2,t1), kz(t1,t2) = kx(t1,t2) + ky(t1,t2) +
kxy(t1,t2) + kxy(t2,t1)
X(t) Y(t) - , kz(t1,t2) = kx(t1,t2) + ky(t1,t2) 2. Y(t)=
X(t)+(t) ( X(t) (t) ) , [Y(t)] =
M[X(t)]+ (t), ky(t1,t2) = kx(t1,t2) 3. Z(t) = (t)X(t) ( X(t) (t)
) , [Z(t)] = (t)M[X(t)], kz(t1,t2) = (t1)(t2)kx(t1,t2)
4. dt
)t(dX)t(Y = ,
dt
)]t(X[dM)]t(Y[M = ,
21
2121
),(2),(
tt
ttkxttk y =
5. =t
0d)(X)t(Y ,
=t
0d)](X[M)]t(Y[M , = d),(kxd)t,t(k 2
2t
021
1t
0121y
11.1 mx(t), Dx(t) kx (t1,t2) X(t) = Ucos2t U- M[U] = 5, D[U] =
6.
: mx(t) = M[u cos2t] = cos2t M[u] = 5cos2t kx(t1,t2) = M[( X(t1)
- mx(t1) ) ( X(t2) - mx(t2) )] = = M[(u cos2t1 5 cos2t1) (u cos2t2
- 5 cos2t2) = M[cos2t1 cos2t2(u 5)
2] = =cos2t1cos2t2 M[(u 5)2]=cos2t1cos2t2
M[(u mu)2] =cos2t1 cos2t2 Du = 6cos2t1 cos2t2
Dx(t) = kx(t,t) = 6cos22t
11.2 mx(t), Dx(t) kx(t1,t2) X(t) =U(3t+1)-5t+4 U - M[U]=3 ,
D[U]=2 : mx(t) = M[u(3t + 1) 5t +4] = (3t +1)M[u] 5t + 4 = (3t + 1)
3 5t + 4 = 4t + 7 kx(t1,t2) = M[(u (3t1 + 1) 5t1 + 4 4t1 7) (u(3t2
+1) 5t2 + 4 - 4t2 7)] = M[(u(3t1 + 1) - 3(3t1 + 1)) (u(3t2 +1) -
3(3t2 +1))]= M[(3t1 + 1)(3t2 +1)(u 3)
2] = =(3t1 + 1) (3t2 +1) M[(u 3)2] = (3t1 + 1) (3t2 +1) M[(u
mu)
2] = = (3t1 + 1) (3t2 +1) Du = 2(3t1 + 1) (3t2 +1) Dx(t) =
kx(t,t) = 2(3t + 1)
2 11.3 X(t) =Vt , V- , [1;4]. mx(t), Dx(t) kx(t1,t2). X(t).
: V
2
5
2
41
2
bamv =
+=
+= ,
4
3
12
1)(4 2
12
a)(b 2Dv =
=
=
[ ] [ ] [ ] t2
5VMttVMX(t)M ===
k x ( t 1 , t 2 ) = M [ ( X ( t 1 ) - mx ( t 1 ) ) ( X ( t 2 ) -
mx ( t 2 ) ) ] =
[ ]
tkD
tttt
ttt
VDVMttttVttVM
xx2
2121
2
212211
4
3),()(
4
3
2
5
2
5
2
5
==
==
=
=
X(t) 1 1V = , 2 2V = , 3 4V = :
t11
X3(t)=4t
X2(t)=2t
X1(t)=t
X(t)
2
4
Copyright & A K-C
-
( ) ( ) 4yy t25t,tktD ==
11.4 X(t) ( ) ( ) 2121xx ttt,t k,4ttm =+= . ( ) ( ) ( )21 t,tyk
tyD ,tym , 2tXt5tY )()( += . : X(t) : 5t 2.
( ) 220252452M[X(t)]5t]2X(t)[5t(t)y ++=++=+=+= ttttMm
,t25t)t,(tk5t5t)t,(tk 22
2121x2121y ==
11.5 X(t) 4tt
21x2
x
22
21
e)t,(tk 5t,t(t)m+
=+= .
8.X(t)t
3Y(t)
2=
:
5t
1585t)(t
38(t)m
38X(t)
3M(t) 2
2x22y=+==
=ttt
m
( )
( ) 2t
4yy
4
tt
2221x2221y
2
22
21
2111
et
9t,tk(t)D
ett
9t,tk
t
3
t
3)t,(tk
+
==
==
11.6 Z(t) -
X(t) Y(t) :
.ett)t,(tk
,tt)t,(tk t,(t)m t,(t)m)t(t
2121y
2121xyx
21+=
===
: Z(t)=X(t)+Y(t), ( ) ( ) ( ) 0tttmtmtm yxz ==+=
( )[ ]21 tt212,1y2,1x2,1z e1tt)tt(k)tt(k)tt(k ++=+= 11.7 2 X(t)
Y(t) : ,t(t)m 2x =
)(2,1
tte)tt(k 211
= , ,1)t(my =
2122 )(
2,1ytte)tt(k
=
, X(t),Y(t) -
tte)tt(k 122,1xy=
: 0)t,t(k 21xy = , tt2t)t(mt)t(m)t(m 22yxz +=++=
( ) ( ) ,,
)()()(
1221
2)12(221
)21(12,1212,12,1
tmettk
e
ztt
xy
yxztttt ttettkttttkttk
+
=
+=+=
tte)tt(ttetttte)t,t(kt
)t,t(kt)t,t(ktt)t,t(k)t,t(k
122
12221121
)(21
)(12xy1
21xy221y2121x21z
+ +++=+
+++=
11.8
tt3)t(m 2x += , )).tt(21(e4)t,t(k 21)tt(221x 21 += - . :
( ) ( ) ( ) 2ttYttXtZ ++=
Copyright & A K-C
-
[ ] [ ] ( )
)1t2t2(tte16t
)t,t(k
t)t,t(k
)tt(tte16t
)t,t(k
tt
)t,t(kt,tk ;1t6)tt3(dt
)t(XdM)t(YM ;
dt
)t(dX)t(Y
21)(2
1
21x
221y
21)(2
1
21x
21
21x2
21y2
21
21
=
=
=
=+=+===
11.9 , dt
)t(dX)t(Y = X(t)
tsin2)t(mx = 2
12 )tt(521x e3)t,t(k = .
. :
)2)12(101()(530
1
)2,1(
2)2,1(
)12()(530
1
)2,1(21
)2,1(2
)2,1(
cos2)sin2())(()(
212
212
ttet
ttk xt
ttk y
ttet
ttk x
tt
ttk xttk y
tttmxtmy
tt
tt
====
====
====
====
============
11.10 X(t) 2
21 )tt(21x
2x e2)t,t(k ,1t)t(m == .
.)t1(dt
)t(dXt2)t(Z ,1t)t(Xt)t(Y 22 +=++=
: 195
[ ] [ ]
)))(21(16),(
4),(
)(4),(
),(22),(
125)1(4)1()1(2)1())((2)(
2),(),(
11)1(1)(1)()(
221
2)21(21
2
21
22121
21
2)21(
1
21
21
212
2121
222222
2)21(21212121
232222
ttttt
ttkt
ttttk
ttt
ttktt
ttkttttk
mm
ttttkttttk
m
tte
tte
ttttttttttt
tte
ttttttttXtMttXtMt
xz
x
xz
xz
xy
y
=
=
=
=
+=+=+=+=
==
++=++=++=++=
11.11 X(t) 5t4)t(mx += 2121x tcostcos)t,t(k = . . :
( ) ( )
( ) tsint,tk)t(Dtsintsindt cost dcosd)t ,(kt d)t,t(k
t5t2d)54(dmtm;d)(X)t(Y
2yy
2120 20 120 21x0 121y
t0
2t0 xy
t0
2121
==
= = =
+=+ == =
11.12
( ) ( ) +=t
tdXt
tY0
2
3
2 . X(t)
.ett)t,t(k ,e2)t(m 21 tt2121xtx +== . :
( ) ( ) ( ) +=+=+= t0
t
0
2t22xy t1e
t3
4tde2
t3
2tdm
t3
2tm
( ) = = + 2t
0
t
0211
2121y ded
t3
2
t3
2t,tk 21
1 2
Copyright & A K-C
-
( )( )( ) ( )2tt
2yy
tt2
tt1
21
1etet9
4t,tk)t(D
1eet1eettt9
42211
+==
++=
11.13 X(t) c
( ) 2121x2x tt5t,tk 3t)t(m =+=
( ) 3t
0tdX)t(Y + =
Y(t): )t(my )t,t(k 21y
: Z(t)=t X(t),
11.14 t2cosVe)t(X t3 = , V- , M[V]=5, D[V]=1. -
, X(t) =t
0d)(X)t(Y .
: X(t)
[ ] [ ][ ]
[ ] [ ]
==
=
===
==
===
+
++
d2cose5d)(m)t(m
t2cost2costte
VDt2cost2costte)5V(Mt2cost2costte
)t2coste5t2costVe)(t2coste5t2costVe(M)t,t(k
t2cose5VtM2coset2cosVeM)t(m
3t
0
t
0xy
21)(3
21)(32
2133
23
23
13
13
21x
t3t3t3x
21
2121
2211
,
3tt2sinU)t(X += , U- , M[U]=2, D[U]=4
: ( ) 3tt2sin2tmx += ; )t,t(k 21x =4 ,t2sint2sin 21 =)t(Dx
t2sin4 2
11.16 X(t): 2
12 )tt(21x
2x e2)t,t(k ,1t2t3)t(m =++= .
3)(2
1)(,1
)()( 2 ++++====++++++++==== ttX
ttZt
dt
tdXttY
: ( ) ,2t2
1t
2
5tmz +=
212 )(
2121z
ttett2
1)t,t(k
=
11.17 X(t) 22
21 tt21x
3x e)t,t(k ,t3t)t(m =+= .
22 t3dt
)t(dXt)t(Y = .
( ) ( ) ( )( ) ( )
( ) ( ) tdZtYtt5t,tkttt,tk
t3t3tttm ttm
3t
0
2221x2121z
32xz
21
+ =
==
+=+==
( ) ( ) ( )
( ) ( ) 33222
0
21
01221
2
0
1
0121
324
3
0
33
0
2121 9
5 5,,
2
3
43
ttdddkdttk
ttt
tdtdmtm
ttt
z
t
y
tt
zy
===
++=++=+=
( )
( ) ( ) ( )13
152cos32sin2
13
53cos32sin2
13
5
2cos32sin213
2cos
30
3
33
+=+=
++=
tteetm
ctte
tdte
tty
tt
Copyright & A K-C
-
( ) ( ) 2121xx tcostcos9t,tk ,tcos2tm: ==
: ,3)( 4ttmy =2
22
1 ttett4)t,t(k 323
121y=
11.18 , =t
0d)(X)t(Y , X(t) -
, X(t) c 198
2121x tcostcos4)t,t(k =
tsin1)t(m 2x += . .
: ,t2sin4
1t
3
2)t(my
= tsintsin4
)t,t(k 21221y =
11.19 X(t): 22212121xx t2ttt)t,t(k,3)t(m ++==
H +=t
01t3d)(X)3t()t(Y
: ,1t6t3)t(m 2y =
( )( )
++= 3122232121y 2211 tt32
tt4
1tt
3
13t3t)t,t(k
11.20 X(t) X(t)=Vcost, V- - mv=2, v=3. X(t): )t(mx , )t,t(k 21x
, xD t( ) . -
Y(t)=X(t)+dt
)t(dX , -
Copyright & A K-C
-
12 C , .
( , ) X(t) - , , , ..
),(k)t,t(k,const)t(m x21xx == tt 12 = , ))(k)(k( xx =
const)0(k)t,t(kD xxx === X(t) , )(Sx , ),( + - ( ) , --:
( ) ( ) ==++
dcosk2
)(S ;dcosS)(k0
xx0
xx
)(Sx , - , , ))(S)(S( xx = -
( ) ( )
( ) )(S2
1)(S ,d ek
2
1)(S
d eSd eS2
1)(k
xx i
xx
ix
ixx
==
= =
+
+
+
)(Sx , )(Sx - , , )(Sx (0; ).
( ) ( ) ===+
+d S
2
1d S)0(kD x
0xxx
)(x )0(k)(k
D
)(k)(x
x
x
xx
==
-
( ) ( )
xx
x SS
S max2
d )(max
1 2x== +
= =++
d)(k 2
d)(20
x2x0
x
1)0(x = ,
)(x , );;( + -
)(Smax x
, )(Sx ,
);( +
12.1 X(t) 1t2)t(mx += .e)t,t(k2
12 )tt(21x
=
12.2 dt
)t(dX)t(Y = ?
:
ttkxtte
ttet
etttt
ttkxttky
constttmxtmy
tt
tttt
12),()2)12(21(
)(2
)12)(2())((
2))((
1221
)2,1(2
)2,1(
2)12())(()(
212
212
212
============
====
++++====
====
====
====++++========
Y(t)- 12.2 X(t)
Copyright & A K-C
-
.e)t,t(k,1)t(m )tt(21xx 21== X(t) Y(t),
Y(t)=t 1dt
)t(dX + ?
: X(t) -
( ) )(kttett)(ttetttttetttdt
)tt(ktt)tt(k
111t1))t(m(t1dt
)t(dXtM)t(m
y)(
212)(
21)(
2121
2,1x2
212,1y
xy
21212
21 ==
=
=
=+=+=
+=
Y(t) , .. )t,t(k 21y
= 2 1t t , t1 t2 12.3 X(t)
,0)t(mx = 221
21x)tt(1
1)t,t(k
+= =
t
0.d)(X)t(Y
X(t),Y(t)? : X(t)- ,
Y(t) , ,
,arctgtt2)t1(n)t,t(kD 2yy ++== l .. t 12.4 X(t)=A cos (t+ ) , A
- , - , [0;2]. , X(t) - :.. [0;2] ,
f( )= [ ]
[ ]
2;0;0
2;0;2
1
, Y=cos Z=sin
M[Y]=M[cos ]= ( ) +
Y f ( )d = 0sin
2
1d
2
1cos
2
0
2
0=
=
M[Z]=M[sin ]= ( ) +
Z ( )d = 0cos
2
1d
2
1sin
2
0
2
0=
=
+= tcos(A[M)t(mx )]= costcosA[M -A
sin t sin ]= =AcostM[cos ]-AsintM[sin ]= 0 0 =0 Y=cos2
Z=sin2
( ) ( )
( )( )
( )
( )( ) ( )
( ) ( ) ( )( )( )
1122
212
22
1221
11
21
1
11
212
112
1
1
0112
2
01
1
0
20122
21
21
0121
0
1
11
2
1
V ;
1 ;
;
;
1 );(
1,
0
211
arctgttarctgtt
tt
ttnttarctgtt
ddV
dduarctgu
VddV
t
dduttarctgu
darctgtarctg
darctgd
dttk
dmtm
t
t tt
t
y
t
xy
++
++
++=
==+
==
==+
===+=
==+
=
==
l
Copyright & A K-C
-
M[Y]=M[cos2 ]= ( ) +
Y f( )d = 0d
2
12cos
2
0=
[ ] [ ] ( ) ( ) 0d2
12sind f Z2sinMZM
2
0=
= ==
+
( ) =+++=++= )tt(cos)2)tt(cos(2
A)tcos(A)tcos(A)t(X)t(X 2121
2
2121
( )[ ]
[ ]
[ ]
2
A)tt(k)t(D
)tt(cos2
A)tt(cos00
2
A
)]tt(cos2sinM)tt(sin]2[cosM)tt([cos2
A
)]tt(cos2sin)tt(sin2cos)tt([cosM2
A
)]t(X)t(X[(M))t(m)t(X))(t(m)t(X(M)t,t(k
)tt(cos2sin)tt(sin2cos)tt(cos2
A
2
,xx
21
2
21
2
212121
2
212121
2
212x21x121x
212121
2
==
=+
=+++=
=+++=
===
+++=
12.5 X(t) )1(e4
b)(k 2x +
= (b>0,
>0).
dt
)t(dX)t(Y = .
: ,
=
=
=
=
==
=
2
x
1
2
x
121
x2
21
21x2
21yy
2
x2
y
td
)(dk
t
t
)(k
ttt
)(k
tt
)t,t(k)t,t(k)(k
d
)(kd)(k
,1t
)tt(
t 1
t
)tt(
t 1
12
12
12
2
=
=
=
=
2x
2
12x
2x
1y
d
)(kd
td
)(kd
d
)(kd
t)(k
=
=
=
>+=
0),1(e4
b
0),1(e4
b
)(k
2
2
x
)1(e4
b
0),1(e4
b
0),1(e4
b
))(k()(k xy =
+
>==
4
b)0(k)t(D yy ==
( )( )( )( )
( )( )
=
>=++=
0,e4
b1e
4
b
0,e4
b1e
4
b
k
2
2
x
( )( )( ) ( )
( ) ( )
=+
>+=
=
0, 1e4
b1e
4
b
0, 1e4
b1e
4
b
k x
Copyright & A K-C
-
( )
>
-
12.10 -
>
-
( ) ( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) )(S)(S)(S2
1)(S)(S
2
1
d cosk2
d cosk2
2
1
dcoscos2
1k
2
0x0x0x0x0x
00
x00
x
000
x
=+=++=
=
++
=
=++=
++
+
.. )(Sx )(S)(S 0x0x =+ 12.13. )(Sx ,
>
=
1 ,0
1 ),1()(k
2
x
:
( ) ( ) ( )
=
+=
==
===
====
+
1
0
10
2
1
0
21
0
2
0
sin1
sin1
)1(2
sin1
V d cosdV
-du 1
d cos12
d cos12
d cos2
)(
du
kxxS
( )2
22
2
2102
2
2
sin4cos1
2cos
2
===
12.14 )(Sx , 22)( = ekx
:
( ) === +
+
deedek iixxS22*
2
1
2
1)( ( ) ( ) =
+=
++
dede2
a
0
i20
i22
( ) ( )
22
2
22
2
2
0
i20i22
4
a2
4
4
2
a
i2
1
i2
1
2
ae
i2
1e
i2
1
2
a
+=
+
=
=
++
=
+
=
++
12.15 , -
= eD)(k xx )0( >
: ( ) ( ) ( ) ( ) =
+
=
+=
==
=
++
+
++
+
+
0
i0ix
0
i0
ix
ix
ix
*x
ei
1e
i
1
2
Ddede
2
D
eeD2
1de)(k
2
1)(S
)(
D
i
1
i
1
2
D22
xx
+=
++
=
.)(
D2)(S2)(S
22
x*xx
+==
=
+==
= D2
)0(D2
)0(S
D2
)(Smax
D2
x
2x
x
x
x
x
( )
=
= == =+
+
+ + 2e2
de2deDD
2d k
D
2000
xx0
xx
.2=
Copyright & A K-C
-
12.16. X(t) 5)t(mx =
)1(
8)(S 2
*x +
= X(t)
:
, = eD)(k xx
)(
D)(S 22x*
x +=
, 1 ,8D 2x ==
8D 1, x == = e8)(kx 12.17 , -
, 0 ,0b,)(
b)(S 22x >>+
=
:
)(2
b)(S
2
1)(S 22x
*x +
==
;2
bD , ;
2
bD x
22x ==
= = e2
b)(k x
( )
=
=
== =
=
==
=
+++
2
2e
2de
2
b
b
4d k
D
2
b
b
)0(S
b
)(SmaxD2
000x
x
2
xx
x
12.18. )(Sx X(t) ,
= coseDk xx :
( )
( ) =+=
===
+
+
+
deeee
deeDdek
iiix
ix
ixx
D
S
2
1
2
cos2
1
2
1)(*
( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
=
++
++
=
=
+++
+
++
+=
=
+++
+=
++
++=
+
++
++
2222x
x
0
ii
0
ii0
ii
0iixiii
0
iii0
x
)(
2
)(
2
4D
i
1
i
1
i
1
i
1
4D
dedede
de4
Ddeeee
deeee4D
))()()((
)(D2222
222x
+++++=
Copyright & A K-C
-
)(S*x )(S*x
-2 2 = = =2, =1 =2, =3 )(S*x 12.19. X(t) 8)t(mx =
42
2*x
625
520)(S
+++
=
X(t) : ,
= coseD)(k xx :
4222222
222x
2222
222x*
x
)(2)(D
))()()((D)(S
+++++
=
=+++
++
=
4222222432223
222222222222
222222
22222
)(2)(2)(2
42)()()2)(()(
)2))((2)((
))()()((
+++=++++
++++++++=
=+++++=
=+++
3
20D
5
22
x
22
=
==+
=
=+
3
522
22
2
82 2
==
=D
x20 , 0> , .. = 2
= 1 , Dx
=20
10
=
, ( ) = cose10k 2x
12.20. )(Sx X(t),
)sin(cose)(k 2x += ( )0, >
: (((( )))) (((( ))))++++
++++
====
++++========
dieediekxSx sincos
22
1
2
1*
Copyright & A K-C
-
(((( )))) (((( )))) (((( ))))
++++
++++
++++====
====
++++++++++++
++++
++++
++++
====
diiei
diiediie
dieieiei
ieiee
dieieieieiee
000
4
20 2
)(2
1
22
1(
0
2
2
(((( )))) (((( )))) (((( ))))
(((( )))) (((( )))) ====
++++
++++++++++++
++++++++
++++
++++++++
++++
++++
dei
dei
dededei
iiii
iiiiii
00
00
0
( )( )
=
+
+
++++
+++
+=
=
++
+
+++
+
++
++
++
+=
))((
)(2
))((
)(2
)(
2
)(
2
4
))(())(()(
1
)(
1
))(()(
1
)(
1
4
22222222
2
2
i
i
i
i
iiiiii
iiiiii
+
++
+
=
=
+
+++
++
+++
+=
2222
2
22222222
2
)(
2
)(
2
2
))(())(()(
1
)(
1
2
12.21. ,
X(t) )0(eD)(k22
xx >= :
( = +
2due 2u2
)
, 22
4x*x e
2
D)(S
= , 2
2
4x*xx e
D)(S2)(S
==
)(Sx
(((( )))) (((( ))))
====
====
====++++====
++++
====
++++
++++
====
====++++
++++
++++++++
====++++
++++
====
====++++
====
++++
====
2
2222
422
4422
2
1
2
1
2
22
22
2
22
4
22
222
222
22
dud
uiu
i
de
i
exD
d
iii
exD
d
i
exD
diexDdiexkxS
22
24
2
24
2
2
2
4
2
22
2222
+
=== eD
eDdu
eeD xx
ux
Copyright & A K-C
-
=
24x
x4
2eD))(S(
2
2
, =0 ))(S( x
+ - =0 - max., 0
( ) ( ) ==
=
= 2
D
D2
Smax
2D ;
D0S
x
x
x
xxx
( )
===
==
=====
+
++
2
22
22
2
dud
2
u 2
22
0
2
2
22
00
due
u
deDD
dkD
u
xx
xx
Copyright & A K-C
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