I i 53 52 53 52 53
) B
1 22 26 , 2 21 27 , B :
P (B H) = 22,
P (B H
) =21
1 48 2 48
:
P (B) =
n
å
i =1
P (H) P (BH) =24 ×22 +29 ×21 =1137 =379
i i 53 48 53 48 2544 848
:
P (A) = 24,
P (B) = 379
53 848
5. I 10+α=29 , 3 ; II 15+ α=34 , 5 . . , . , I ?
n 1 = 29 - 1
a 1 = 3 - 1
b 1 = 26 - 1
n 2 = 34 -
2
a 2 = 5 -
2
b 2 = 29 -
2
- { }
:
P ( 1 A)
:
H 1 - H 2 -
{ 1- }
{ 2 - }
:
P (H) = 1,
P (H
) = 1
1 2 2 2
n
:
P (H) = P (H) + P (H
) = 1 + 1 = 1 ()
å i
i =1
1 2 2 2
:
ì ,
,ü
A H 1 - í
1- ý
î þ
ì , ,ü
A H 2 - í
2 - ý
î þ
P (A H) = 26,
P (A H
) =29
1 29 2 34
A:
n
P (A) =
å
i =1
P (H) P (A H) = 1 × 26 + 1 × 29 = 1725
I i 2 29 2 34 1972
1- , , , :
Times;26
P ( A) = = 2 29 =
0,51
:
P (A)
P ( A) =
1 1725
6. (3+α)=22 , , 2 . 3 . . . .
n = 22 -
a = 2 - b = 20 -
|
|
X -
:
X,
F (x)
X . 0, 1, 2.
:
20!
C 0 × C 3 1×
1140 57
P (X
=0) = 2 20 = 3!17! = =
C 3 22! 1540 77
22
3!19!
20!
C 1 × C 2 2 ×
380 19
P (X
=1) = 2 20 = 2!18! = =
|
C 2 × C 1
1540 1540 77
1× 20 1
P (X = 2) = 2 20 = =
|
( ) X:
xi | 0 | 1 | 2 |
pi | 57/77 | 19/77 | 1/77 |
:
n
å pi
= P (X = 0) + P (X = 1) + P (X = 2) = 57+ 19+ 1
= 1 ()
i =1
77 77 77
X<x, X , , x
X, . .
F () = P (X < ).
1) x ≤ 0, ;
F (x) =0, .. X,
x =0
2) 0 < £1,
F ()= F (1)= P (X <1)=57/ 77, ..
£1 X
X =0;
3) 1 < £2, X 0 1.
F () = F (2) = P (X < 2) = 57 / 77 +19 / 77 = 76 / 77;
4) >2, X 0; 1; 2
F ()= P (X < x) =57/ 77 +19 / 77 +1/77=1.
ì0,
x £ 0
|
F (x) =
0 < x £ 1
|
1 < x £ 2
ïî1,
2 < x
F (x)
ì0,
x £ 0
|
: F (x) =
0 < x £ 1
|
1 < x £ 2
ïî1,
2 < x
xi | 0 | 1 | 2 |
pi | 57/77 | 19/77 | 1/77 |
7. X Y :
- Z=X+Y : (X+Y)=M(X) + M(Y)
zij = xi + y j
z 1 = 2 -1 = 1
z 2 = 2 + 0 = 2
z 3 = 2 + 20 = 22
z 4 = 4 -1 = 3
z 5 = 4 + 0 = 4
z 6 = 4 + 20 = 24
:
p 1= P (Z =1)= P (X p 2 = P (Z =2) = P (X
=2) × P (Y =-1) =0,7 ×0,4 =0,28
=2) × P (Y =0) =0,7 ×0,1 =0,07
p 3= P (Z =22) = P (X =2) × P (Y =20) =0,7 ×0,5=0,35
p 4 = P (Z =3)= P (X p 5 = P (Z =4) = P (X
=4) × P (Y =-1) =0,3×0,4 =0,12
=4) × P (Y =0) =0,3×0,1 =0,03
p 6= P (Z =24) = P (X =4) × P (Y =20) =0,3×0,5=0,15
Z , :
zi | 1 | 2 | 3 | 4 | 22 | 24 |
pi | 0,28 | 0,07 | 0,12 | 0,03 | 0,35 | 0,15 |
|
|
n
M (X) = å xi × pi
i =1
t
M (Y) = å y j × pj
j =1
= 2 × 0, 7 + 4 × 0, 3 = 2, 6
= -1× 0, 4 + 0 × 0,1 + 20 × 0, 5 = 9, 6
r
M (Z) = M (X + Y) = å zk × pk
k =1
= 1× 0, 28 + 2 × 0, 07 + 3 × 0,12 + 4 × 0, 03 + 22 × 0, 35 +
+24 × 0,15 = 12, 2
12, 2 = 2, 6 + 9, 6 Û M (X + Y) = M (X) + M (Y) ()
:
zi | 1 | 2 | 3 | 4 | 22 | 24 |
pi | 0,28 | 0,07 | 0,12 | 0,03 | 0,35 | 0,15 |
8. :
ì 0,
ï
x < 0
|
ï
x, 0 £ x £ a = 19 19
îï 1,
x > a = 19
, , 0,3+α=19,3, 0,7+α=19,7. .
ì 0,
ï
x < 0
|
ï19
îï 1,
x > 19
:
P (19,3 < X < 19, 7),
f (x),
D (X)
X , :
b
P (a < X < b) = ò f (x) dx = F (b) - F (a)
a
P (19, 3 < X < 19, 7) = F (19, 7) - F (19, 3) = 1-1 = 0
, :
ì 0,
ï
x < 0
ì0, x Î(-¥;0)È(19;+¥)
|
ï19
ïî 0,
x > 19
îï 19
, x Î[0;19]
X .
+¥
M (X) =ò x × f (x) dx.
-¥
M (X) = ò x ×
dx = = = 9,5
+¥
-¥
19 3 19 3
1
D (X) = ò x 2 ×
dx - 9,52 = x
- 90, 25 = 19 - 90, 25 = 361
57 12
ì0, x Î(-¥;0) È (19; +¥)
: P (19,3 < X < 19, 7) = 0,
f (x) = ï 1
, D (X) = 361
í, x Î[0;19] 12
îï 19
9. .
a / 40 =19 / 40 = 0, 475.
- 3- . 3- .
p = 0, 475
X -
3 -
:
X,
M (X),
D (X)
X . 0, 1, 2, 3. , :
p = P (X
= m) = Cmpmqn - m, q = 1 - p,
m = 0, 1, 2,3.
m n
P (X
P (X
P (X
P (X
= 0) = C 0× 0, 4750 × 0,5253 = 1×1× 0,5253 = 0,145
= 1) = C 1× 0, 4751× 0,5252 = 3× 0, 4751× 0,5252 = 0,393
|
|
= 2) = C 2 × 0, 4752 × 0,5251 = 3 × 0, 4752 × 0,525 = 0,355
= 3) = C 3× 0, 4753× 0,5250 = 1× 0, 4753×1 = 0,107
( ) X:
xi | 0 | 1 | 2 | 3 |
pi | 0,145 | 0,393 | 0,355 | 0,107 |
, X, , :
M (X) = np,
D (X) = npq,
M (X) = 3× 0, 475 = 1, 425,
:
D (X) = 3× 0, 475 × 0,525 = 0,748
xi | 0 | 1 | 2 | 3 |
pi | 0,145 | 0,393 | 0,355 | 0,107 |
M (X) = 1, 425, D (X) = 0,748
10. α=19 . , . . , : ) 3 ; ) 4 6 . .. ... .
q = 0,19 - , X - ,
Y -
:
) P (X
> 3),
) P (4 £ X
£ 6),
M (Y),
D (Y)
X . 0, 1, 2, 3,,m,. , :
pm = P (X
= m) = qmp, q = 1- p,
m = 0, 1, 2,3,..., m,....
) P (X
> 3) = 1- P (X £ 3) = 1- P (X
= 0) - P (X
= 1) - P (X
= 2) - P (X = 3) =
times; 0,81- 0,191× 0,81- 0,192 × 0,81- 0,193 × 0,81 0, 00130
) P (4 £ X £ 6) = P (X
= 4) + P (X
= 5) + P (X
= 6) =
= 0,194 × 0,81+ 0,195 × 0,81+ 0,196 × 0,81 0, 00129
Y , X, 1 (.. 1, 2, 3,,n,), :
M (Y) = 1,
D (Y) =
q,
p p 2
M (Y) =
D (Y) =
0,81
0,19
0,812
1, 235
0, 290
: ) P (X
> 3) = 0, 00130, ) P (4 £ X £ 6) = 0, 00129,
M (Y) = 1, 235,
D (Y) = 0, 290
11. α=19 . , , 2 : ) 2 ; ) 1 . .. ... 1 .
l = 19 - ( )
X - 2 Y - 1
:
) P (X < 2),
) P (X ³ 1),
M (Y),
s (Y)
, X
.
: a = l ×D t =19× 2 = 38. ,
, :
P (X < 2) = P (X = 0) + P (X = 1); P (X ³1) =1- P (X = 0).
:
P (X = m) = a
× e - a.
|
0 1
P (X = 0) = a
× e - a = e -38;
|
|
P (X = 1) = a
× e - a = 38 e -38
0! 1!
P (X < 2) = e -38 + 38 e -38= 39 1, 22×10-15;
e 38
P (X ³ 1) = 1- e -38 1
Y
a * = l ×D t * =19×1 =19. :
M (Y) = a * = 19; D (Y) = a * =19; s (Y) =
: ) P (X < 2) = 39,
e 38
) P (X ³ 1) = 1-
1,
e 38
M (Y) = 19, s (Y) =
12. 0,002. , 800+10α=990 .
p = 0, 002
n = 990
X -
: P (X £ 2)
, , (i - ) , :
m
P (X = m) = a
m!
× e - a, a = np,
a = 990 × 0, 002 = 1, 98
0 1 2
P (X £ 2) = P (X = 0) + P (X = 1) + P (X = 2) = a
× e - a + a
× e - a + a
× e - a =
1 é a 2ù 1 é
1, 982 ù
0! 1! 2!
= × 1+ a + = × 1+1, 98 + 0, 682
ea ê 2 ú e 1,98 ê 2 ú
ë û ë û
: P (X £ 2) = 0, 682
13. . , ) α+10=29 ; ) α+5=24 60 . .. ... .
X
X -
:
a) P (0£ X £29),
) P (24£ X £60),
M (X),
s (X)
X :
ì 0,
ï -
x < a
ì 0,
ï
x < 0
|
b - a
a £ x £ b Û F (x) = ï x, 0 £ x £ 50
|
ï ï
îï 1,
x > b
îï 1,
x > 100
,
P (0£ X £29) = F (29) - F (0) =29 -0 =0,58
P (24£ X £60) = F (60) - F (24) =1-24 =0,52
:
a + b
M (X) =
,
, s (X) = =
2
M (X) = 0 +50 = 25
(50 - 0)2
s (X) = =
12 3
:
) P (0£ X £29) =0,58,
) P (24£ X £60) =0,52,
M (X) = 25,
s (X) = 25 3
3
14. α=19 . 9 . , , 3 4 ( ) : ) α+2=21 ; ) α+1=20 α+3=22 . .. 10 .
l = 1
:
a) P (T ³21),
) P (20£ T £22),
M (T),
s (T)
1 , λ = 1/19 /.
) P(T ³ t) , t :
P (T ³ t) = e - lt, t ³0
t = 21 :
- 1 ×21
P (T ³21) = e 19
0,331, t ³ 0
) , a b :
P (a £ £ b) = e - la - e - lb,
- 1 ×20
. .
- 1 ×22
P (20£ £22) = e 19
- e 19
0, 035
: 10 4 19 ( 9.00, 9.19, 9.38, 9.57), 10 , .. 10.16 . , τ = 16 . :
M () = lt =
1 ×16 = 16,
19 19
: D () = lt =16
: ) P (T ³21)0,331,
) P (20£ £22)0,035,
|
|
M () =16,
D () =16
19 19
15. 2 a = 38
a = 19. , ) ;
) -1 3;
) 2.
X
M (X) = a = 38
s = 19
:
) P (X < 0)
) P (-1 £ X ) P (X - a
£ 3)
£ 2)
) (α;β) :
æ b - a ö æ a - a ö
P (a < X < b) = 0ç s
÷ - 0 ç s
÷,
x z 2
è ø è ø
0 (x) =
1 -
ò e 2 dz -
,
.
P (X < 0) = P (-¥ < X < 0) =
æ 0 -38 ö +0,5=
0 ç 19 ÷
è ø
=- 0(2) +0,5=-0, 4773 +0,5=0,0227
) :
P (a < X < b) = P (a £ X £ b) = P (a < X £ b) = P (a £ X < b),
P (-1 £ X £ 3) =
æ 3 -38 ö -
æ -1-38 ö
0 ç 19 ÷ 0 ç 19 ÷
è ø è ø
0(-1,8421) - 0(-2,0526) =-0,4673 +0,4799 =0,0127
) (a-ε; a+ε), a, :
æ e ö
P (X - a
< e) = 2 ç ÷,
|
P (X - a
£ 2) = 2
æ2 ö 2 × 0, 0419 = 0, 0838
:
0 ç 19 ÷
) P (X < 0) = 0, 0227,
|
) P (X - a
£ 2) = 0, 0838
16. , a +100 =119
10 a + 200 = 390 . , , , , ... . , a + 200 = 219 .
X
P (119 £ X £ 390) 0,9973
:
M (X) = a, s,
P (X
> 219)
:
P (X - a < 3 s) = P (-3 s + a < X < 3 s + a) 0,9973
P (119 £ X £ 390) = P (119 < X < 390) 0,9973
, :
|
|
í í
2 a = 509 ì
|
- í
a = 254,5
î3 s + a = 390
(390 a)
ï s =
271
ï s = 45,17
î 3 î 6
, :
æ b - a ö æ a - a ö
P (a < X < b) = 0ç s
÷ - 0 ç s
÷,
P (X
è ø è ø
> 219) = P (219 < X < +¥) = 0,5 -
æ ö
ç219 - 254,5 ÷
0 ç 271 ÷
ç ÷
è 6 ø
0,5+ 0(0,7860) =0,5+0,2841 =0,7841
: M (X) = a = 254,5,
s =271,
6
P (X
> 219) = 0,7841
17. 15 X. . α =19 :
; | ; | ; | ; | ; | ; | ; | ; | ; | ; | ||||||||||
; | ; | ; | ; | . |
:
1)
2) Mo Me
3)
4)
5)
6)
1) . ( ) X
; | ; | ; | ; | ; | ; | ; | ; | ; | ; | ||||||||||
; | ; | ; | ; | . |
2) Mo Me:
.
Mo =23, .. , 5 .
, ( ) .
, :
NMe
= n +1, n .
2
NMe
=15 +1 =8,
2
Me = 23, .. 8 , 23.
3) .
. -
xi
(),
mi.
xi | ||||||
mi |
4)
:
F ˆ (x) = mx
n
, mx
,
n - . : n =1+3+3+5 +2 +1=15.
:
ì0,
x £ 20
|
20 < x £ 21
ï4 / 15,
|
ï12 / 15,
ï
ï14 / 15,
ï
21 < x £ 22
22 < x £ 23
23 < x £ 24
24 < x £ 25
î1,
x > 25
5) .
,
n
å xi mi
x = i =1,
n
xi
- ().
( MS Excel)
/ | xi | mi | ximi | (xi - x) mi |
6,0844 | ||||
6,4533 | ||||
0,6533 | ||||
1,4222 | ||||
4,7022 | ||||
6,4178 | ||||
∑ | - | 25,7333 |
n
å ximi 337
x = i =1 = 22, 47
n 15
6) . . .
n
å i i
(x - x)2 m
D = i =1 = 25, 7333 1, 7156 -
()
,
n 15
s = = 1, 3098
S 2 =
n D =
15 ×1, 7156 1,8381 -
( )
n -1 15 -1
, S = = 1, 3558
18. 30 X. . α =19 :
; | ; | 20,5 | ; | ; | 21,5 | ; | ; | ; | ; | 19,5 | ; | ; | |||||||
; | ; | 19,5 | ; | 24,5 | ; | 23,5 | ; | ; | ; | ; | 22,5 | ; | ; | ||||||
20,5 | ; | ; | ; | ; | ; | ; | ; | ; | ; | 21,5 | . |
:
1) , ;
2) Mo Me;
3) , ;
4) ;
5) ;
6) (.. ) .
1) ,
; | ; | ; | ; | 19,5 | ; | 19,5 | ; | ; | 20,5 | ; | 20,5 | ; | ; | ||||||
; | 21,5 | ; | 21,5 | ; | ; | ; | ; | ; | ; | ; | ; | ||||||||
22,5 | ; | ; | ; | ; | 23,5 | ; | ; | ; | 24,5 | ; | ; | . |
(.. ):
R = x max - x min = 25 -19 = 6
() :
m1+ 3,322lg n =1+3,322lg30 5,91,
n .
m=6 ( ). :
h = R = 6 = 1
m 6
(xi-xi+1) | , mi | , Si | |
19-20 | |||
20-21 | |||
21-22 | |||
22-23 | |||
23-24 | |||
24-25 | |||
∑ | - | - |
2)
(xi-xi+1) | , mi | Pi= mi/n | mi/h | Pi/h | , Si | |
19-20 | 6/30 | 6/30 | ||||
20-21 | 3/30 | 3/30 | ||||
21-22 | 4/30 | 4/30 | ||||
22-23 | 8/30 | 8/30 | ||||
23-24 | 4/30 | 4/30 | ||||
24-25 | 5/30 | 5/30 | ||||
∑ | - | - |
, 8, 22-23, .. :
Mo = xMo
+ h ×
PMo - PMo -1, (PMo - PMo -1) +(PMo - PMo +1)
xMo =22 ;
h = 1 ;
: 2016-11-24; !; : 6723 |
:
, .
==> ...