1. .
1. 8 , = 0,1. , , .
( ) .
X n 0,1,2,, n. . , . , , . (1- ).
P (X =0) = (1- )n.
(n -1)
(1- )n.
n , .
P (X =1) = n (1- )n.
(n 2). n2 n 2, 2(1- )n-2.
P (X =2) = n2 2 (1- )n-2.
P (X =k) = nk 2 (1- )n-k. (1)
,
nk = n!/(k!(n-k)!).
. : n = 8, = 0,1
P (X =0) = (1-0,1)8 = 0,430;
P (X =1) = 8∙0,1∙(1-0,1)7 = 0,383;
8∙7∙6∙5∙4∙3∙2∙
P (X =2) = ---------------- (0,1)2(1- 0,1)6 = 0,149;
2∙6∙5∙4∙3∙2∙
P (X >=2) = 1 - P (X =0) - P (X = 1) = 0,187.
2. n = 1000 , = 0,001. , , .
, n = a/n a
P (X =k) = ak/k! e-a. (2)
(2) , k n. k, n.
a = n = 1, n . (2)
P (X =0) = e -1 = 0,368;
P (X =1) = e -1 = 0,368;
P (X =2) = ½ e -1 = 0,184;
P (X> 1) = 1-0,368-0,368 = 0,264;
2. ( ) . (.. . . , 1970, .405.).
|
|
() m τ.
α , m m 1 m 2
P (m 1 < m < m 2) = α.
[ m 1 , m 2] . m 1, m 2 m , α 1, .
, m , τ , m τ 1, τ 2,, τN N
m = (τ 1 + τ 2++ τN)/ N. (1)
.
1. m (1).
2. 0,8; 0,9; 0,95; 0.99
( , ).
3. m 1 , m 2 ( ), , m ( -). , -.
n < 15 m 1 , m 2
m 1 = ν 1 m , m 2 = ν 2 m , (2)
ν 1 = 2 N /χ2δ, ν 2 = 2 N /χ21-δ. (3)
χ2δ χ21-δ χ2- . , . p r ( ). χ2 (. .. , .. , , 1988).
r = 2 N, (4)
χ2δ p = δ, χ21 -δ p = 1- δ. δ α
δ = (1 α)/2. (5)
1. 10
i | ||||||||||
τ i, |
m 0,9.
(1) N = 10
m =4250/10 = 425 .
(4) (5)
r = 20, δ = 0,05; 1- δ = 0,95.
χ2δ = 31,4; χ21 -δ = 10,85.
(2) (3)
ν 1 = 20/31,4 = 0,637; ν 2 = 20/10,85 = 1,84;
m 1 = 425∙0,637 = 271 ; m 2 = 425∙1,84 = 783.
, 0,9 m 271 783 .█
|
|
0,9 [271,784] m. 0,9 .
.
2. α = 0,8; 0,6, 1.
: m 1 = 299 , m 2 = 684;
m 1 = 340 , m 2 = 582. █
:
α | m 1, | m 2, |
0,6 | ||
0,8 | ||
0,9 | ||
0,96 | ||
0,98 |
. .
α=0,96; 0,98 , . 551.
n > 15 χ2 ,
m 1 =4 m N /(√(4 N -1) +ε0)2, m 2 =4 m N /(√(4 N -1) - ε0)2, (6)
ε0
α/ 2 = Φ(ε0), (7)
Φ(ε0) = , .
ε0 ( , ), Φ(ε0). ε0 α/ 2 , .
3. 25 1600 . m α= 0,8.
m = 1600/25 = 64 . α/ 2 =0,4 Φ(ε0) = 0.39973 ε0 = 1,28.
(6)
m 1 =4∙1600/(√99 +1,28)2=50,5 ; m 2 =4∙1600/(√99 - 1,28)2=84 .█
n .
4. m = 64 . α= 0,8, , 250, 2500, 25000 .
: m 1 = 59,2 , m 2 = 69,5.
m 1 = 62,4 , m 2 = 65,7.
m 1 = 63,5 , m 2 = 64,5.
m 1, m 2 N (.) █