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Slides \ 076_Otrbin.tex

.
1. . N BI N 2. . = { = (, )T , > 0, 0 < < 1} 3. . N B I N (, ) NT . 4. . 4.1. . L(N B I N (, )) = N B I N (r; , ),
r

N B I N (r; , ) =
j =0

nbin(j ; , ), r 0.

4.1.1. N B I N (r; , ) = 1 - N B I N (r; , ) =

nbin(j, , ),
j =r+1

N BI N (r - 1; , ) = BE T A(1 - ; r, ) = 1 - B I N (r - 1; + r - 1, 1 - ). 4.1.2. N BI N = (r; , ) = B E T A( ; , r + 1) 4.1.3. N B I N , BI N BE T A (. 4.1.1, 4.1.2, .....). 4.2. . L (N B I N (, )) = nbin(r; , ), nbin(r; , ) = , nbin(r; , ) = C .. 4.2.1. nbin(r + 1; ; ) ( + r) = (1 - ), nbin(r; , ) (r + 1) r = 0, 1, . . .
r +r-1

( + r) (1 - )r , ( )r!

r = 0, 1, 2 . . . .

(1 - )r ,

r = 0, 1, 2, . . . .

4.2.2. . ( - 1) 1- , ( - 1) 1- rmod = (1- )-1 0, 4.2.3. r

( - 1)

1-

1-

, ( - 1)
1- 1 < .

-

nbin(r; , ) ; r nbin(r; , ) . 5. . . , N B I N . + N B I N (, ) - , N B I N (, ) - . = 1 , . ( , ) N BI N . 6. . 6.1. N B I N (1 , ) N B I N (2 , ) , N B I N (1 , ) + N B I N (2 , ) = N B I N (1 + 2 , ). 6.1.1. , Z1 , . . . Z ... , Z1 = GE OM ( ), N B I N (, ) =
i=1 d d d

Z

i

6.2. . , / = P OI S (t), = G(, ). = N B I N (, ), = , = 1 . 1 + t
d d d

6.3. . y1 , . . . , y ... , y1 = LOG( ),
d

0 < < 1,

, = P OI S ().

y1 = N B I N
i=1

d

-

,1 - . ln(1 - )

6.4. , N B I N (, ) - 6.5. ,
1- 1- 2

1-

= N (0, 1) + 0d (1)

d

0,

1-

< ,

N B I N (, ) = P OI S () + 0p (1). 7. . 7.1. . (t) = 1 - (1 - ) exp{it}

.

7.2. . µ(t) = 7.3. . (t) = 7.4. . (t) = 1 - (1 - )e-
t

1 - (1 - )e

t

,

t < ln

1 . 1-

1 - (1 - )z

,

|z | <

1 . 1-

,

t 0.

7.5. .

7.5.1. . k (t) = ln - ln(1 - (1 - ) exp t). 8. . , n = - 1 = 1- .

8.1. . 1- a1 = , (1 - ) {1 a2 = 2 (1 - ) a3 = {1 3 (1 - ) a4 = {1 4 + 3 (1 - )3 } 8.2. . m m m
2

+ (1 - )}, + (1 + 3 )(1 - ) + 2 (1 - )2 }, + (4 + 7 )(1 - ) + (1 + 4 + 6 2 )(1 - )2 + .

= = =

(1 - ) , 2 (1 - ) (2 - ), 3 (1 - ) {1 + (4 + 3 )(1 - ) + (1 - )2 }. 4 1- , 1-
2

3

4

8.3. . f f f
1

= = =

2

( + 1)

, 1-
3

3

( + 1)( + 2)

.
4

1- f4 = ( + 1)( + 2)( + 3) ... ... .............................. 1-

,

k

f

k

=

( + 1) · · · ( + k - 1)

,

k = 1, 2 . . . .

8.4. . k k k k kr
1

= = = = =

2

3

4

(, ) +1

1- , (1 - ) , 2 (1 - ) (2 - 3 (1 - ) {1 + 4 kr (, (1 - ) (1 -

), 4(1 - ) + (1 - )2 }, ) . )

8.5. . 0 =

8.6. . 1 = 8.7. . 2 =

2- (1 - )

.

2 + 6(1 - ) . (1 - )

8.8. . 8.8.1. . b1 = E |N B I N (, ) = m =
(1- ) 1

2m( + m - 1)! -1 (1 - )m , m!( - 1)!

+ 1 = [a1 + 1]. 2 = m2 1 =. a1

8.8.2. .

9. . 9.1. . 9.1.1. , .. . I 0 ( ) = 2 . (1 - ) 9.1.2. , , I 0 ( , ) 10. . 10.1. N B I N , n , n = 1, 2, . . . , t; N B I N , n
n

= (t; N B I N (, )).

10.2. "" - P{N B I N (, ) m} = P{B I N (m + - 1, ) m}. 10.2.1.
+m-1

P{N B I N (, ) m} =
k =m

b(k ; + m - 1, ) =
m-1

( + m - 1)! = ( - 1)!(m - 1)!
0

v

(1 - v ) -1 v .

10.3. N B I N < 1 , > 1 ( ) = 1 , .. . 10.4. N B I N (1 , ) N B I N (2 , ) , N B I N (1 , ) + N B I N (2 , ) = N B I N (1 + 2 , ).

10.5. , 1 ,

1

-1 . N B I N (, ) = P OI S (0) + 0d (1).

10.6. , = const . ....... 10.7. . 10.7.1. sin h
-1

N B I N (, ) .

10.7.2. Anscomb (1948). - 0.5 sin h
-1

N B I N (, ) + 3/8 . - 3/4

11. . 12. . 12.1. . 12.1.1. . y = (y1 , . . . , yn ) , yi N , yi ... , i = 1, n , y1 = N B I N (, ). S =
n i=1 d

yi , S = N B I N (n, ).
d

12.1.2. . (y(1) , . . . y(n) ) . 12.2. . 12.2.1. . n = n = ^0

n , S + n

n - 1 S + n - 1 . b( ) = E {n - n } = E ^0 S (S + n )(S + n - 1) > 0.