στατιστικη 1- Πετροπουλος

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url: http://www.aegean.gr

83200

Copyright ,

All rights reserved
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1

-

1.1.

. X

=(X1, X2,...,Xn)

fX

(x

, ), ,

. .

, , ,

g() : Rk,k 1, . X

. Xi, i = 1, 2, . . . , n

, , X
http://www.math.aegean.gr/http://lastpage/http://fullscreen/http://close/http://quit/http://quit/http://close/http://fullscreen/http://lastpage/http://www.math.aegean.gr/

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.

1.1.1

1.1.2 (

g(), g() : Rk) ( g()). 1.1.3 T = T(X

) -

g(),

ET(X

) = g(), .

T(X

), (T, ), ,

1.1.4 (T, ) = E(T(X

) g())2

1.1.5 (T, ) = Var(T(X

)) + (E(T(X

)) g())2. b(T, ) =E(T(X

)) g() T -

g(), ,

(T, ) = Var(T(X

)) + b2(T, ).

1.1.6 T g(),

(T, ) = Var(T(X

)).

1.1.7 T1 T2

g(), ,

(T1, )(T2, ),

(T1, 0)< (T2, 0), 0

.

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1.1.8 T1 T2( -

) g(), T2

g().

1.1.9 T ( )

g(),

g().

, , .

1.1.10 X1, X2, . . . , X n

f1(x, ),

g() = ,

X = 1

n

ni=1

Xi, .

1.1.11 X1, X2, . . . , X n f1(x, ), g() = 2 , S2 =

1

n 1n

i=1

(Xi X)2

2.

1.2.

1.2.1 X1, X2, . . . , X n U(0, ), =(0, ), . -

1.2.2 X1, X2, . . . , X n U(0, ), =(0,

), 2. -
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1.2.3 2

. 1.2.1 T = T(X

) = 2 X

, T2

2

. ,

ET2

=VarT+(ET)2

= 2

3n+2 =

3n+ 1

3n 2 E(T2) = 3n+ 1

3n 2 E

3n

3n+ 1T2

=

2, ,

3n

3n+ 1T2 2.

2, -

1.2.2,

g().

1.2.4 X P(), = (0, ), 2. -

1.2.5 X P(), = (0, ), g() = e

k

k!k = 0, 1, . . .. -

1.2.6 X1, X2, . . . , X n ,

Xi B(ni, ), i = 1, 2, . . . , n . - . -

1.2.7 , T1 = T1(X

) T2 = T2(X

)

, , -

T =a1T1 + a2T2 . -

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1.2.8 X

= (X1, X2, . . . , X n) Y

=(Y1, Y2, . . . , Y n)

R 21

,

22

.

1. , Tc =Tc(X

, Y

) = cX + (1 c)Y, c R, .

2. Tc, ,

.

-

1.2.9 X, Ber-

noulli = (0, 1). g() = 2; -

1.2.10 X ,

fX(x; ) =

(1 )x , x {0, 1, . . .}, (0, 1)0 ,

.

,

T(X) =

1 , X = 00 , X 0. . . ; -

1.2.11 X

= (X1, X2, . . . , X n)

2, ,

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1.

ni=1

aiXi

ai

ni=1 ai

= 1.

2.

ni=1

ai = 1, ai Var

ni=1

aiXi

. - ;

-

1.2.12 X

Poisson , = (0, ). . -

1.2.13 T g(),

S = aT +b

ag() + b. f, f(T)

f(g()); -

1.2.14 X = (X1, X2) B(1, ),

(0, 1). 2 C ={T(x1, x2) :T(0, 0) = 0, T(1, 1) = 1, T(1, 0) =T(0, 1)}. -

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2

2.1.

, (. 1.1.9)

.

2.1.1 T = T(X

)

() g(),

1. T , . ET =g(),.

2. VarT VarT1, T1 g().

2.1.1 -

,

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.

Cramer-Rao

.

(1) R.

(2) S ={x

:fX

(x

; )> 0} .

(3)

Rn

fX

(x

; )dx

=

Rn

fX

(x

; )dx

,.

(4)

Rn

T(x

)

fX

(x

; )dx

=

Rn

T(x

)fX

(x

; )dx

, -T(X

).

(5) I() = E

lnfX

(x

; )

2

, 0< I()

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1. I() =E

2

2 lnfX

(x

; )

, .

2. X

=(X1, X2, . . . , X n) ,

Xi ,fXi(xi; ),i = 1, 2, . . . , n ,

I() =

ni=1

Ii()

Ii() = E

lnfXi(xi; )

2.

3. X

=(X1, X2, . . . , X n) ,

I() = nI1()

I1() Fisher

X1, X2, . . . , X n.

Cramer - Rao

(1) - (5), X

().

2.1.3 {fX

(x

; ), } () ,

1. S ={x

;fX

(x

; )> 0} .

2. fX

(x

; ) = eA()+B(x

)+c()D(x

),x

S,.

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2.1.4 X

=(X1, X2, . . . , X n) -

fX

(x

; ) c()( fX

(x

; ))

, (2), (3) (4) -Cramer-Rao (4) T =T(X ).

, ,

g() .

2.1.5 X

=(X1, X2, . . . , X n) -

fX

(x

; ) fX

(x

; ) = eA()+B(x

)+c()D(x

)

,

a) R.

b)c() .c) 0< I()

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, fX

(x

; ) = eA()+B(x

)+c()T(x

),x

S, , X

.

2.1.7 2.1.5 2.1.6 g()

Cramer-Rao X

g() ,g() = ED(X

)

ED(X

).

2.1.7

Cramer-Rao -

, ,

. ,

.

2.1.8 X

= (X1, X2, . . . , X n)

fX

(x

; ), , T(X

)

X

T(X

) = t t

.

, ,

,

Neyman-Fisher.

2.1.9 T(X

) fX

(x

; ) =

q(T(x

); )h(x

),x

, qh .

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2.1.10 -

.

1. X

=(X1, X2, . . . , X n) .

2. T(X

) = (X(1), X(2), . . . , X (n)) , X(i), i =

1, 2, . . . , n .

3. T1 = T1(X

) T2 = K(T1) = K(T1(X

)),

K()1 1, T2(X

) .

,

.

2.1.11

(.

).

2.1.12 , g() -

.

-

.

2.1.13 (Rao-Blackwell) T = T(X

)

S = S(X

) g(). S = E(S|T).,

1. S .

2. ES

=ES, , S g(), S g().

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3. VarS VarS, S

T, S =S.

4. (S, )

(S, ),

S

T, S =S.

, S g() -

T, S

S =E(S|T) SRao-Blackwell Rao-Blackwell - S.

2.1.14 T1 T2 S

g(). S1 =E(S|T1) Rao-Blackwell S, T1 S

2

=E(S|T2) Rao-Blackwell S, T2. , 2.1.13 .

.

2.1.15 T = T(X

) , -

,

E(T) = 0, (t) = 0 tT, (T) = 0.

, T, 0

, T .

2.1.16 (Lehmann-Scheffe) T = T(X

) -

S g(). S =E(S|T) g().

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Lehmann-Scheffe ,

, . ,

, , Lehmann-Scheffe.

2.1.17 T = T(X

)

S g(),

T. S g().

,

, X

() .

2.1.18 {fX

(x

; ), } (), k,

1. S ={x

;fX

(x

; )> 0} .

2. fX

(x

; ) = eA()+B(x

)+k

j=1cj()Dj(x

),x

S,.

2.1.19 1 .

2.1.20 X

= (X1, X2, . . . , X n)

k, ,

1. T(X

) =(D1(X

), D2(X

), . . . , D k(X

)) .

2. (c1(), c2(), . . . , c k()) -

Rk, T(X

) .

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, 2.1.20 2.1.17

. , Basu,

, ,

( ).

2.1.21 T(X

) S(X

)

, .

T(X

)S(X

) .

, , , ,

2.1.21 .

2.1.22 X1, X2, . . . , X n,n 2 N(, 2), ,

1. X , (X1 X , X2 X , . . . , Xn X) ..

2. X , S2 = 1

n 1n

i=1

(Xi X)2 ..

3.

ni=1

(Xi X)2

2 2n1.

2.2.

2.2.1 X1, X2, . . . , X n, n 2 -N(, ), = (0, ),

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Fisher Cramer-Rao

1/2. -

2.2.2 X1, X2, . . . , X n, n

2 Poisson

P(ai), ai, i = 1, 2, . . . , n > 0, Fisher, C.R.-..

g() = . -

2.2.3 X1, X2, . . . , X k,k 2 B(ni, ), ni, i = 1, 2, . . . , k = (0, 1), g() = Cramer - Rao. -

2.2.4 X

= (X1, X2, . . . , X n),n

2

Gamma(a, ), a > 0 = (0, ) , Fisher C.R.-..

2.

Cramer - Rao; -

2.2.5 X

= (X1, X2) Bernoulli

B(1, ), =(0, 1), T =T(X

) = X1 + X2

, S =S(X

) =X1 . -

2.2.6 X1, X2 X1B(1, ) X2 B(1, 2), = (0, 1/2). T =T(X

) = X1 + X2 . -

2.2.7 X

= (X1, X2, . . . , X n), n 2 - U(0, ), = (0, ). - . -

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2.2.8 X

=(X1, X2, . . . , X n),n 2 , fX(x; ) = e

(x), x , R, . -

2.2.9 X

= (X1, X2, . . . , X n), n 2 - U(, ), = (0, ). . -

2.2.10 X

={1, 0, 1}.

.. X

1 2 3-1 1/6 1/6 2/3

0 1/3 1/3 1/3

1 2/3 1/6 1/6

T(X) = X . -

2.2.11 S T1 = 1(S)

T2 = 2(S) ,

S. T1 =T2. -

2.2.12 X

= (X1, X2) , X1 N(, 1) X2 N(2, 1), T(X

) = (X1, X2) .

-

2.2.13 X

= (X1, X2, . . . , X n), n 2, - N(, 2), = (0, ).

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T(X

) =

ni=1

Xi,

ni=1

X2i

, .-

2.2.14 X

= (X1, X2, X3) Ber-

noulliB(1, ), = (0, 1).1. .

2.

S1 =

1 , X1 = 1

0 ,

,

S2 =

1 , X1 = 1, X2 = 10 , 2.

3. , 2 g() = P(X1X2).-

2.2.15 X

= (X1, X2, . . . , X n) -

N(, 2), R > 0. k, k > 0. -

2.2.16 X

= (X1, X2, . . . , X n)

U(0, ), > 0.

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1. r,r > 0.

2. ,

(...), cX(n), c .

-

2.2.17 X1, X2, . . . , X n ,

Xk Gamma(k, ), > 0. r,r >n(n+ 1)

2. -

2.2.18 X

=

{0, 1

}.

.. X

-1 2

0 2/3 1/3

1 1/3 2/3

. -

2.2.19 X1, X2, . . . , X

n , Xi P(ti), > 0 ti > 0, i = 1, 2, . . . , n . 2. -

2.2.20 X1, X2, . . . , X n -

U(0, ), > 0. E( 2n

n+ 1X|X(n)) = X(n). -
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3

3.1.

,

( ),

. .

,

X

=x

,

= (x

) x

.

3.1.1 X

= (X1, X2, . . . , X n)

fX

(x

; ), , ( )

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,

L() = L(|x

) = fX

(x

; ).

3.1.2 =

(X

), L() =max

L()

(...) .

3.1.3 ...

, . ln x x, L()

ln L(). .

3.1.4 1.

= (1, 2, . . . , k).

2. ,

L()

.

3.

, -

L() ...

3.1.5 ...

1. 3.1.2 ( )

.

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2. ... ,

.

3. = (X

) ... , ... g()

g().

4. ... ( ) (. 3.1.6)

3.1.6 Tn =T(X1, X2, . . . , X n),n = 1, 2, . . . -

g(). Tn

limn

P(|Tn g()|> ) = 0, > 0.

g()

.

3.1.7 Tn ,

1. VarTn 0, n +.2. b(Tn, ) = ETn g() 0, n +.

Tn g().

3.1.8 ... ( ) -. X1, X2, . . . , X n

f1(x, ) ... , ,

1. (n +) ,

N(, 1I()

)

I() Fisher.

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2. , -

, Sn, N(, 2

n()), (

),2n() 1

I().

...

, ... g(),

.

.

3.1.9 X1, X2, . . . , X n

f1(x, ), =(1, 2, . . . , r),

k =1

n

ni=1

Xki

k.

3.1.10 ,

1. k k -

mk =EXk.

2. k mk,

,k mk.

3.1.11

(...) :

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1) r , r, m1, m2, . . . , m r,

1, 2, . . . , r,

m1 =m1(1, 2, . . . , r), . . . , m r =mr(1, 2, . . . , r).

2)

mi(1, 2, . . . , r) = i , i = 1, 2, . . . , r .

, 1,2, . . . ,

r,

1, 2, . . . , r.

3.2.

3.2.1 X1, X2, . . . , X n ,

Xk P(k), =(0, ), ... 2. -

3.2.2 X1, X2, . . . , X n N(, 2), R > 0, ... , 2. -

3.2.3 X1, X2, . . . , X n U[1, 2],1, 2 R1 < 2, ...

1. 1, 2 ,

2. 2, 1 .

-

3.2.4 X1, X2, . . . , X n -

Ge(), [0, 1]. ... . -

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3.2.5 X1, X2, . . . , X n -

f(x; ) =e(x)

1 + e(x)

2

, x R = R. ... . -

3.2.6 X1, X2, . . . , X n -

U[, ]. ... . -

3.2.7 X1, X2, . . . , X n

U[2, 3]. ... . -

3.2.8 X1, X2, . . . , X n

E(), > 0.

1. ... g() = P(X1 > t),

t .

2. ... .

3. .

-

3.2.9 X -

,

.. X

1 2 3

-1 1/2 1/2 0

0 1/6 1/6 2/3

1 1/3 1/3 1/3

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... ={1, 0, 1}. -

3.2.10 ( ) -

, . 10

,

1540, 1620, 1570, 1552, 1605, 1580, 1590, 1557, 1563, 1547.

,

, i) 1700 , ii) 2000 iii) 2500

-

3.2.11 X1, X2, . . . , X n Ray-

leigh, f(x; ) =x

ex

2

/2, x > 0 > 0.

EX2

= 2 EX4

= 82.

1. ... .

2. .

3. ... ;

-

3.2.12 X1, X2, . . . , X n

E(, ), f(x; , ) = 1

e(x)

, x R > 0.

1. ... .

2. .

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3. ... .

-

3.2.13 X Bernoulli B(1

, ), [ 13

, 23

].

1. ... .

2. ... ...

1 = 1

2.

-

3.2.14 (X1, X2) X1 N(0, 2) X2N(0, 2/4), > 0 . ... g() =P(X

2

1 + 4X2

2 > a), a > 0. -

3.2.15 X1, X2, . . . , X n

U[, ]. . -

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4

Bayes minimax

4.1.

, -

. , ..,

, -

.

, .

,

.

, (),

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, ,

(i)() 0, (ii)

()d = 1 (

() = 1).

() , -

(. ) .

L(t, ) -

R(T, ) = EL(T(X

), ).

, ,

, , ,

, ,

BR(T) = ER(T, ) =

R(T, )()d

Bayes T.

Bayes,

Bayes.

4.1.1 T = T(X

) Bayesg(),

L(t, ) (),

R(T, )()d

R(T, )()d

T =T(X

).

, Bayes

(|x

) =f(x

; )()

f(x

)(4.1)

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f(x

) =

f(x

; )()d.

.

4.1.2 , , -

(|x

),

, , .

Bayes.

4.1.3 X

= x

Bayes T = T(X

) -

g() L(t, ) () T(x

) = t, t t h(t) =

L(t, )(|x

)d.

, , , L(t, ) =

(tg())2, Bayes , .

4.1.4 g() - L(t, ) = (tg())2. X

= x

BayesT = T(X

)

g() T(x

) = Eg(Y), Y -

(|x

).

T = T(X

) ,

max

R(T, ) . .

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4.1.5 T = T(X

) minimaxg(),

L(t, ),

max

R(T, )

max

R(T, )

T =T(X

).

minimax .

4.1.6 T = T(X

) Bayes -

g(), L(t, ) ()

, ,

R(T, ) =c , ,

T =T(X

) minimaxg().

4.2.

4.2.1 X

= (X1, X2), X1

P()

X2 P(2) () = e, > 0.

1. .

2. Bayes g1() = g2() =

2, .

-

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4.2.2 X1, X2, . . . , X n Bernoulli

B(1, ), (0, 1). U(0, 1),

1. Bayes, L(t; ) =(t

g())2

(1 ) .2. minimax , .

-

4.2.3 X1, X2, . . . , X n , XiP(i), i = 1, . . . , n , > 0. ,() = e, > 0 ,

Bayes . - 4.2.4 X1, X2, . . . , X n -

, f1(x; ) = ex, > 0.

, () Gamma(a, ), a, > 0 -, Bayes g1() = g2() = e

L(t, ) = 2(t g())2. - 4.2.5 X1, X2, . . . , X n

Xi N

(ti, 1), tiR , i = 1, 2, . . . , n

R.

N(a, 2) L(t, ) = (t g())2, Bayes . - 4.2.6 X

= (X1, X2, . . . , X n) U(0, ), > 0

. , () = 1, 0< < 1 -

L(t, ) = (t g())2

2 , Bayes

g1() = g2() = 2. -

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.

34 240

4.2.7 X1, X2, . . . , X n -

, f1(x; ) =

x+1, > 0 x > 1.

, () = e, > 0

L(t, )=

(t g())2

, Bayes . -

4.2.8 Bayes

. -

4.2.9 X1, X2, . . . , X n Bernoulli

B(1, ), (0, 1). Beta(a, ), 0 < < 1 a, > 0, minimax ,

. -

4.2.10 X1, X2 Bernoulli B(1, ), (0, 1). L(t, ) =

(t )2(1 ) . T(X

) =1

2

minimax . -

4.2.11 minimax, ,

minimax. -

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. . .

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.

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5

5.1.

-

g() , , x

,

t = T(x

) g(), T(X

)

. , -

, g(), . -

,

g().

, ,

g(). , .

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. . .

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.

36 240

5.1.1 T1 = T1(X

) T2 = T2(X

) T1 < T2,

[T1(X

), T2(X

)] (..)

g() (..) 100(1 a)%,

P(T1(X )g()T2(X )) =

1 a , . 1a ,

a, ,

, () 100a% -

g().

, ( -

)

a, [T1, T2] P(T1

g()

T2)

1a. T1 T2 P(T1g()T2) - 1 a. , , n .

5.1.2 T1 = T1(X

) T2 = T2(X

) T1 < T2,

[T1(X

), T2(X

)] ( )

(...) g()

(..) 100(1 a)%, lim

n

P(T1(X

)g()T2(X

))

= 1 a , .

, ,

g(). g()

,

.

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. . .

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.

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5.1.3 T1 =T1(X

)T2 =T2(X

) . ,

P(T1(X

)g()) = 1 a , ,

T1

(X ) (..) g() (..)100(1 a)%. , P(g()T2(X

)) = 1 a , ,

T2(X

) (..) g() (..)

100(1 a)%. -

.

5.1.4 T1 =T1(X

)T2 =T2(X

) . ,

limn

P(T1(X

)g())

= 1 a , ,

T1(X

) (...) g()

(..) 100(1 a)%. ,

limn P(g()T2(X

)) = 1 a , , T2(X

) (...) g()

(..) 100(1 a)%. , .., ..

g().

, -

.

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.

38 240

, ..

, ,

. , ,

,

, [T1, T2] ,

P(g()< T1(X

)) = P(g()> T2(X

)) =a

2.

,

() .

5.1.5 .. -

g() .

1. T = T(X

, ) ,

. ..

(pivotal quantity) .

2. c1 < c2, T

a, ,

P(c1T(X , )c2)= 1 a,

,

0

< a < 1

.

3. () -

,

P(T1(X

)g()T2(X

)) = 1 a , ,

T1(X

), T2(X

).

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. . .

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.

39 240

4. [T1(X

), T2(X

)] .. g(). . 100(1 a)%.

() ,

, .

5.1.6 X1, X2, . . . , X n ()

F(x),. .

1. .. Yi =F(Xi),i = 1, 2, . . . , n , U(0, 1).2. .. Zi =2 ln F(Xi),i = 1, 2, . . . , n , X22.

3. .. T =

2

n

i=1

ln F(Xi),

X2

2n

X

=(X1, X2, . . . , X n).

5.1.7 T

() ,

, ()

.

5.2.

5.2.1 X1, X2, . . . , X n, XkN( +k, 1), k = 1, 2, . . . , n , R . .. T =

n

X n+ 1

2

,

( ) .. . . 1 a(0< a < 1). -

5/19/2018 1-

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40 240

5.2.2 X1, X2, . . . , X n

E(), > 0 .

1. .. T = 2

n

i=1

Xi .

2. .. .. 100(1 a)%.3. .. .. 100(1 a)%.

-

5.2.3 X,

fX(x; ) =

2

xex2

, x > 0 > 0 .

1. .. T =X2

.

2. .. . . 100(1a)% g1() = g2() = P(X > 1).

-

5.2.4 X1, X2, . . . , X n -

f1(x; ) =e(x), x R .

1. .. .. 100(1 a)%.

2. .. .. 100(1 a)%.-

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.

41 240

5.2.5 X1, X2, . . . , X n -

U(0, ), > 0 .

1. .. .. 100(1 a)%.

2. .. .. 100(1 a)%.

3. .. .. .. 100(1 a)%.

-

5.2.6 X1, X2, . . . , X n -

U(, ), > 0 . ..

2

. . 100

(1 a)%. -

5.2.7 X , ..

fX(x; ) =2

1 x

, 0 < x < .

.. .. 1 a. -

5.2.8 X1, X2, . . . , X n Beta(, 1),

> 0 . ..

.. 1 a. - 5.2.9 n,

90% .. N(, 1) 1/5. -

5.2.10 , X 8.2n

N(, 25). -

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5.2.11 X1, X2, . . . , X n Gamma(2, ),

> 0 ..

0, c

ni=1

Xi

g() = 2.

c, .. 1

a.

(0< a < 1) -

5.2.12 X E(), c, [0, cX] .. .. 100(1 a)%. -

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.

43 240

6

6.1.

1.

X B(n, p), ,

fX(x) =P(X =x) = n

x px(1 p)nx,x {0, 1, . . . , n }, 0< p < 1.

EX =np, VarX =np(1 p). n = 1, .. X Bernoulli.

2. Poisson

X P(), ,

fX(x) =P(X =x) =e

x

x!, x {0, 1, . . .}, > 0.

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.

44 240

EX =, VarX =.

3.

X NB(r, p), ,

fX(x) =P(X =x) =

r + x 1x

pr(1 p)x, x {0, 1, . . .}, 0< p < 1.

EX =r(1 p)

p ,VarX =

r(1 p)p2

.

r = 1, .. X (X Ge(p)).4.

X N

(, 2), ,

fX(x) =1

2e

(x)222 ,x R, R, > 0.

EX =, VarX =2.

= 0 = 1, .. X (X N(0, 1)).X N(, 2) X

N(0, 1).

X

N(, 2)

aX +

N(a+ , a22).

5.

X Gamma(a, ), ,fX(x) =

1

(a)axa1 e

x , x > 0,a > 0, > 0.

(a) =

+

0

xa1 exdx.

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. . .

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.

45240

EX =a, VarX =a2.

a = 1, .. X (X E()). a = r

2 = 2, .. X r

(X X2

r).

X N(, 2)X

2 X2

1.

X Gamma(n, ) 2

X X22n,n Z+.

6.

X

U(a, ), ,

fX(x) = 1

a,x(a, ),a < .

EX =a+

2,VarX =

( a)212

.

7.

X Beta(a, ), ,

fX(x) =1

B(a, ) xa1 (1 x)1,x (0, 1),a > 0, > 0.

B(a, ) = (a) ()(a+ )

=

0

1

xa1 (1 x)1dx.

EX = a

a+ ,VarX =

a

(a+ )2(a+ + 1).

a == 1, .. X U(0, 1).

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.

46 240

8.

X E(, ), ,fX(x) =

1

e

x , x, R, > 0.

EX =+ ,VarX =2.

X E(, )X E().

6.2.

6.2.1 X fX,

x. , ,

h : R R, Y = h(X) . S ={x R|fX(x)> 0}T S h. ,1. h :S T , -

x =h1(y),yT.2. h1 T,

Y,fY, ,

fY(y) = fX(h1(y)) ddyh1(y) .

6.2.2 k- X

= (X1, . . . , X k),

fX

= fX1 ,...,Xk,

x

= (x1, . . . , x k). , , h : Rk Rk, y

=

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.

47240

h(x

) = (h1(x1), . . . , h k(xk)), Y

= h(X

) k- .

S={x

Rk|fX

(x

)> 0}T S h. ,

1. h :S T , -x =h

1

(y

) =(g1(y), . . . , g k(y)),yT.2. h1,

gji(y

) =

yigj(y1, . . . , y k), i, j = 1, 2, . . . , k

T.

.. Y

,

fY

(y

) =fX

(h1(y

))|J|

J ,

J =

g11 g12 . . . g 1kg21 g22 . . . g 2k

gk1 gk2 . . . g kk

.

6.3.

X1, X2, . . . , X k .

1. Xi B(ni, p),i = 1, 2, . . . , k k

i=1

Xi B(k

i=1

ni, p).

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. . .

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.

48 240

ni = 1,i = 1, 2, . . . , k , Xi B(1, p)k

i=1

Xi B(k, p).

2. Xi P(i), i = 1

,2

, . . . , k

ki=1 Xi P(

ki=1 i).

3. Xi N(i, 2i), i = 1, 2, . . . , k k

i=1

Xi N(k

i=1

i,

ki=1

2i).

4. Xi Gamma(ai, ), i = 1, 2, . . . , k k

i=1

XiGamma(k

i=1

ai, ).

ai = 1,i = 1, 2, . . . , k , Xi E()k

i=1

Xi Gamma(k, ).

ai = ri2 , = 2i = 1, 2, . . . , k , Xi X2rik

i=1

Xi X2ki=1ri

.

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. . .

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.

49 240

: 1.1.10.

1.2.1

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. . .

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.

50 240

: 1.1.11.

1.2.2

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. . .

Bayes . . .

.

51 240

: 1.1.10.

1.2.4

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. . .

Bayes . . .

.

52 240

: 1.1.3 .

1.2.5

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. . .

Bayes . . .

.

53 240

: 1.1.3 .

1.2.6

1 1 3

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. . .

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.

54 240

: 1.1.3 .

1.2.7

1 1 3

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. . .

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.

55240

: 1.1.3 .

1.2.8

: 1 1 3

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. . .

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.

56 240

: 1.1.3 -

.

1.2.9

: 1 1 3

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. . .

Bayes . . .

.

57240

: 1.1.3

.

1.2.10

: 1 1 3

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. . .

Bayes . . .

.

58 240

: 1.1.3

, Lagrange.

1.2.11

: 1.1.3

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. . .

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.

59 240

: . .3

e.

1.2.12

: 1.2.4

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. . .

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.

60 240

1.2.13

: 1.1.3 .

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. . .

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.

61 240

1.2.14

: -b

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.

62 240

(a, b) (, X U(a, b)) EX = a + b2

, ,

2. , 1.1.10,

, X , ,

EX =

2E(2 X) = , ,

T(X

) = 2 X .

1.1.6, T(X

),

. ,

VarT =Var(2 X) = Var 2n

n

i=1

Xi = 2n

2

Var n

i=1

Xi =4

n2

ni=1

VarXi =4

n2

ni=1

2

12=

2

3n.

, (T, ) =Var(T) =2

3n.

1.2.1

: -2

5/19/2018 1-

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. . .

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.

63 240

(a, b) (, X U(a, b)) VarX = (a b)2

12, ,

2

12. , 1.1.11,

, S2 ,

ES2

=2

12E(12S2) = 2, ,

T(X

) = 12S2 2.

1.2.2

: Poisson

P( )

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. . .

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.

64 240

(, X P()) EX = VarX = , , . , X,

2, X2.

E(X2) = VarX + (EX)

2= + 2 = EX +

2

E(X

2) = EX + 2

E(X

2)

EX =

2 E(X2 X) = 2 E(X(X 1)) = 2, . , T(X) = X(X 1) 2.

1.2.4

:

()

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.

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g() ,

g() = P(X = k), .

T(X) =

1 , X =k0 , X k. .,

E(T(X)) = 1 P(T(X) = 1) + 0 P(T(X) = 0) = P(T(X) = 1) =P(X =k) = e

k

k!,.

T(X) g().

1.2.5

: X1, X2, . . . , X n , -

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. . .

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.

66 240

T =T(X

) =

n

i

=1

Xi B(n

i

=1

ni, ).

,

ET =(

ni=1

ni)E

1

ni=1

ni

T

=, .

, T1 =T1(X

) =

ni=1

Xi

ni=1

ni

.

, 1.1.6,

(T1, ) = Var(T1(X

)) = Var

ni=1

Xi

ni=1

ni

=

1

ni=1

ni

2

Var(

n

i=1

Xi) (T1, ) =

(1 )n

i=1

ni

.

1.2.6

: T

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. . .

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.

67240

ET(X

) =, .

, ET =

E(a1T1 + a2T2) =

a1ET1 + a2ET2 =

a1 + a2 =

, a1 +a2 = 1.

1.2.7

:

5/19/2018 1-

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. . .

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.

68 240

1. Tc , ETc =,.,

ETc =E(cX + (1

c)Y) = cE(X) + (1

c)E(Y) =c + (1

c) = ,

.

, Tc

c R.2. , Tc

, c

( ) Tc.

VarTc =Var(cX + (1 c)Y) = c2Var(X) + (1 c)2Var(Y) =c22

1

n + (1 c)2

2

2

n =

21

+ 22

n c2 2

22

n c+

22

n .

Tc c (

ac2 + bc+ d),

c0 = b2a

= 22

2

n

22

1+2

2

n

=2

2

21

+ 22

.

,

,

Tc0 =2

2

21

+ 22

X +2

1

21

+ 22

Y .

1.2.8

: T =T(X) 2.

X Bernoulli, ,

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.

69 240

X Bernoulli, ,

P(X =x) = x(1 )1x, x {0, 1}.

,ET =

2 T(0)P(X = 0) + T(1)P(X = 1) =2 T(0)(1 ) + T(1) =2

2 + (T(0) T(1)) T(0) = 0 (6.1)

(6.1) (0, 1), . - 2.

1.2.9

: X ,

fX (x; ) =P(X =x) =(1 )x. ,

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.

70 240

fX( ; ) ( ) ( ) ,

ET(X) = 1 P(T(X) = 1) + 0 P(T(X) = 0) = P(T(X) = 1) =P(X = 0) =,(0, 1),

T(X) .

, ,

(0, 1), T(X) (0 1),

,

.

1.2.10

:

n

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.

71 240

1.

ni=1

aiXi ,

E n

i=1

aiXi =, .

,

E

ni=1

aiXi

= ni=1

aiE(Xi) = n

i=1

ai =n

i=1

ai = 1.

2. Var n

i=1

aiXi n

i=1

ai =

1, Lagrange,

,

L =L(a1, . . . , a n, ) = Var

ni=1

aiXi

n

i=1

ai 1

=

ni=1

a2iVarXi

ni=1

ai 1

=

ni=1

a2i2

ni=1

ai 1

=2n

i=1

a2i n

i=1

ai 1.

, L, a1, . . . , a n -

n+ 1 n+ 1.

L

a1= 2a1

2 = 0 (1)

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.

72 240

L

a2= 2a2

2 = 0 (2)

. . .L

an= 2an

2 = 0 (n)

L

=

ni=1

ai 1 = 0 (n+ 1)

(1) (n), a1, . . . , a n ,

a1 =a2 =... = an =

22 (6.2)

(n+ 1),

ni=1

22 1 = 0 = 2

2

n . (6.3)

, (6.2) ,

a1 =a2 =... =an =

1

n

, T(X

) =1

n

ni=1

Xi,

. ,

( -

) .

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.

73 240

1.2.11

: T(X) ,

E (T (X )) (0 )

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.

74 240

E(T(X)) = , (0, ).

,

E(T(X)) = x=0

T(x) x

x!e =

x=0

T(x) x

x!=e.

e ,

x=0

T(x)x

x!=

x=0

x

x!

x=0

T(x)

x!x =

x=0

x+1

x!

x=0

T(x)

x!x =

x=1

x

(x 1)!

T(0) +

x=1

T(x)

x!x =

x=1

1

(x 1)! x, (0, ).

,

T(0) = 0 T(x)

x!=

1

(x 1)! , x = 1, 2, . . .

, T(X) = X ( -

).

1.2.12

: T g(),

,

E T ()

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.

75240

ET =g(), .,

ES =

E(aT +

b)=

aET +

b =

ag()+

b,

., .

-

. 1.2.4, T(X) = X

, f() = 2,

f(T(X)) = X2 , T1(X) =X(X 1). , T g(), f(T)

f(g()), f .

1.2.13

: T = T(X1, X2) 2,

(0, 1),1 1

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.

76 240

ET =2

1x1=0

1x2 =0

T(x1, x2)P(X1 =x1, X2 =x2) =2

T(0

,0

)P(X1 = 0

, X2 = 0

)+

T(0

,1

)P(X1 = 0

, X2 = 1

)+

T(1, 0)P(X1 = 1, X2 = 0) + T(1, 1)P(X1 = 1, X2 = 1) = 2

T(0, 0)P(X1 = 0)P(X1 = 0) + T(0, 1)P(X1 = 0)P(X1 = 1)+

T(1, 0)P(X1 = 1)P(X1 = 0) + T(1, 1)P(X1 = 1)P(X1 = 1) = 2

T(0, 0)(1 )2 + T(0, 1)(1 ) + T(0, 1)(1 ) + T(1, 1)2 =2

{T(0, 0)T(0, 1)T(1, 0) + T(1, 1)}2 + {T(0, 1) + T(1, 0)2T(0, 0)} + T(0, 0) = 2. (6.4)

(6.4) (0, 1) ,T(0, 0) T(0, 1) T(1, 0) + T(1, 1) = 1T(0, 1) + T(1, 0) 2T(0, 0) = 0T(0, 0) = 0

T(1, 1) = 1

T(0, 1) =T(1, 0)T(0, 0) = 0

2 ,

C ={T(x1, x2) :T(0, 0) = 0, T(1, 1) = 1, T(1, 0) =T(0, 1)}.

1.2.14

: 2.1.4

Cramer-Rao.

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.

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2.2.1

: 2.1.4

Cramer-Rao.

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. . .

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.

78 240

2.2.2

: 2.1.5 .

5/19/2018 1-

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. . .

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.

79 240

2.2.3

: 2.1.4 -

Cramer-Rao 2.1.5 .

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. . .

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.

80 240

2.2.4

: 2.1.9 T

2.1.8 S .

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. . .

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.

81 240

2.2.5

: 2.1.8 T -

.

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.

82 240

2.2.6

: 2.1.9

2.1.15 .

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. . .

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.

83 240

2.2.7

: 2.1.9

2.1.15 .

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. . .

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.

84 240

2.2.8

: |Xi|, i = 1, 2, . . . , n 2.2.7

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. . .

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.

85240

2.2.9

: 2.1.10(1)

2.1.15 .

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. . .

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.

86 240

2.2.10

: 2.1.15.

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. . .

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.

87240

2.2.11

: 2.1.15 .

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. . .

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.

88 240

2.2.12

: 2.1.9

2.1.15 .

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. . .

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.

89 240

2.2.13

: 2.1.16.

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. . .

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.

90 240

2.2.14

: 2.1.17.

5/19/2018 1-

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. . .

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.

91 240

2.2.15

: 2.1.17.

5/19/2018 1-

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. . .

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.

92 240

2.2.16

: 2.1.17.

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. . .

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.

93 240

2.2.17

: 2.1.10(1)

2.1.15 .

2.1.17 .

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. . .

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.

94 240

2.2.18

: 2.1.20 -

2.1.17 .

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. . .

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.

95240

2.2.19

: 2.2.16 2.1.16.

5/19/2018 1-

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. . .

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.

96 240

2.2.20

: C.R.-..

Cramer-Rao, (1) - (5).

(1) =(0, ) R.(2) - (4) ,

2 1 4

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. . .

Bayes . . .

.

97240

2.1.4. .

1. S ={x :f(x , )> 0} = Rn+ .

2. f(x

, ) =

ni=1

121/2

e(xi)2

2 = 1

(2)n/2n/2 e

(xi)2

2

f(x

, ) = exp

n2 ln(2) n2 ln 12n

i=1

(xi )2.

,

A() =n

2 ln(2) n

2 ln , B(x ) = 0,c() = 1

2 D(x ) =

ni=1

(xi )2

.

.

c() = 12

,

c() = 1

22 (0, ), (2)(4), 2.1.4.

(5) Fisher.

,

I() = nI1(), I1() =E

2

2 lnf(x; )

.

,

f(x; ) =1

21/2e

(x)22 lnf(x; ) =1

2ln(2) 1

2ln (x )

2

2

lnf(x; ) =

12

+(x )2

22

2

2 lnf(x; ) =

1

22 (x )

2

3 .

, I1() =E3

(X )2

2

= 3

E(X )2

2

=

3

VarX

2

= 3

2

=

1

22.

,I() = nI1() =n

22 > 0 = (0, ).

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. . .

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.

98 240

( ) 1( )22

( )

Cramer-Rao , C.R.-..

g() = 1/2 ,

(g())2

I()=

((1/2)1/2)2

n/(22)=

2n.

2.2.1

: 2.2.1 (1)

- (5) Cramer - Rao (1),

2.1.4 (5).

(1) =(0, ) R.(2) - (4) Poisson

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. . .

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.

99 240

(2) - (4) Poisson ,

1. S ={x

:f(x

, )> 0} = Rn+ .

2. f(x

, ) =

ni=1

eai(ai)

xi

xi!=e

ai

ni=1

axii

xi 1ni=1xi!

f(x

, ) = exp

(

ni=1

ai) n

i=1

ln xi! n

i=1

xiln ai + ln

ni=1

xi

,

A() =(n

i=1

ai), B(x

) =n

i=1

ln xi! n

i=1

xiln ai, c() =ln

D(x

) =

ni=1

xi.

.

c() = ln , c() = 1

(0, ), (2)(4), 2.1.4.(5) Fisher.

,

I() =

ni=1

Ii(), Ii() =E

2

2 lnf(xi; )

.

,

f(xi; ) = eai(ai)

xi

xi! lnf(xi; ) =ai + xiln ai + xiln ln xi!

lnf(xi; ) =

ai + xi

2

2 lnf(xi; ) = xi

2 Ii() =E

Xi

2

Ii() = ai

.

Fisher I () =ni=1ai > 0 (0 )

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.

100 240

, FisherI() = > 0 (0, ). Cramer-Rao , C.R.-..

g() = ,

(g())2

I()=

1

(n

i=1ai)/ =

ni=1ai

.

2.2.2

: .

1. S ={x

:f(x

; )> 0} ={0, 1} . . . {0, 1} ={0, 1}k .

2. f(x, ) =

k ni

xi(1 )nixi =

k ni

xi(1 )

ni

xi

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.

101 240

f (

, ) i=1 xi ( ) i=1 xi ( )= exp

k

i=1

ln

ni

xi

+ (ln )

ki=1

xi + ln(1 ) k

i=1

nik

i=1

xi

= exp

k

i=1

ln

ni

xi

+ ln(1 )

ki=1

ni + ln

1 k

i=1

xi

. ,

A() = ln(1)k

i=1ni, B(x ) =

ki=1

ln ni

xi

, c() = ln

1 D(x ) =k

i=1xi,

.

2.1.5,

a) =(0, 1) R.

b)c() = ln

1

,c() =1

(1 ) ,

(0, 1).

c) , Fisher ,

I() =

ki=1

Ii(), Ii() =E

2

2 lnf(xi; )

.

,f(xi; ) =

ni

xi

xi(1 )nixi

lnf(xi; ) = ln

nixi

+ (ln )xi + (ln(1 ))ni (ln(1 ))xi

lnf(xi; ) = xi

ni xi

1 2

2 lnf(xi; ) =

xi

2 ni xi

(1 )2 .,

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.

102 240

Ii() =E Xi2 niXi(1 )2 = EXi2 + ni EXi(1 )2 = ni2 + ni ni(1 )2Ii() =

ni

(1 ) .

,I() =

ki=1

ni

(1 ) .

, D(X

) =

k

i=1

Xi g() =

ED(X

) =E

ki=1

Xi

= ki=1

ni

,

ki=1

Xi

k

i=1

ni

.

2.2.3

: C.R.-..

2 (1) - (5) Cramer-

Rao.

(1) =(0, ) R.(2) - (4) Gamma(a, ) .

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. . .

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.

103 240

1. S ={x

:f(x

; )> 0} = Rn+ .

2. f(x

, ) =

ni=1

1

(a)axa1i e

xi =

1

((a))nna

ni=1

xi

a1

e

xi

= exp

nln((a)) naln + (a 1)

ni=1

ln xi 1

ni=1

xi

.

A() =nln((a)) naln ,B(x ) =(a 1)n

i=1ln xi,c() =

1

D(x

) =

ni=1

xi .

(5) , Fisher

,

I() =nI1(), I1() =E 2

2 lnf(x; ) .

,

f(x; ) =1

(a)axa1e

x

lnf(x; ) = ln((a)) aln + (a 1) ln xx

lnf(x; ) =a

+x

2

2

2 lnf(x; ) =

a

2 2 x

3

I1() =E

a

2 2X

3

= 2

EX

3 a

2 = 2

a

3 a

2 =

a

2.

, (0, ) FisherI() = na2

-

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. . .

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.

104 240

.

C.R.-..

g1() = . ,

I1 =(g

1())2

I()=

2

na

C.R.-.. g2() = 2.

,

I2 =(g

2())2

I()

= 44

na

.

, Cramer-Rao.

2.1.5, -

Cramer-Rao, D(X

) =

ni=1

Xi

g() = ED(X

) = E

n

i=1 Xi =

n

i=1 EXi = na , , Cramer-Rao.

2.2.4

: X1 X2 , f(x

; ) = f(x1; )f(x2; )

f(x

; ) = x1 (1 )1x1 x2 (1 )1x2 =x1 +x2 (1 )2(x1+x2). Neyman-Fisher ( 2.1.9),

h(x

) = 1 q(T(x

), ) = x1 +x2 (1)2(x1+x2), T(X

) = X1 +X2 .

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. . .

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.

105 240

S , , s S ,

X

|S = s . s = 1 ,

P(X1 = 1, X2 = 0|X1 = 1) = P(X1 = 1, X2 = 0, X1 = 1)P(X1 = 1)

= P(X1 = 1, X2 = 0)

P(X1 = 1)=

P(X1 = 1)P(X2 = 0)

P(X1 = 1)=

P(X2 = 0) = 1 . , S .

2.2.5

: 2.2.5,

tT , X

|T =t . t = 1 ,

P(X1 = 1, X2 = 0|X1 + X2 = 1) = P(X1 = 1, X2 = 0, X1 + X2 = 1)P(X1 + X2 = 1)

=

P(X 1 X 0) P(X 1)P(X 0)

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. . .

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.

106 240

P(X1 = 1, X2 = 0)

P(X1 + X2 = 1) =

P(X1 = 1)P(X2 = 0)

P(X1 = 1)P(X2 = 0) + P(X1 = 0)P(X2 = 1) =(1 )2

(1 )2 + (1 2) =2 23 4 , -

T =T(X

) = X1 + X2 .

6.3.1 2.2.5 -

, T =T(X

) =X1 + X2 ,

.

2.2.6

: ,

f(x

; ) =

ni=1

f(xi; ) =

ni=1

1

I(0,)(xi) =

1

n

ni=1

I(0,)(xi).

X(n) =max{X1, X2, . . . , X n} -X(1) =min{X1, X2, . . . , X n}, ,

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. . .

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.

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ni=1

I(0,)(xi) =1 , 0< xi < , i = 1, 2, . . . , n 0 , . =

1 , 0< x(1)x(n) < 0 , . n

i=1

I(0,)(xi) =I(0,x(n)](x(1))I[x(1),)(x(n))

,f(x

; ) =1

nI(0,x(n)](x(1))I[x(1),)(x(n)).

, h(x

) = I(0,x(n)](x(1)) q(T(x

), ) = 1

nI[x(1),)(x(n)),

Neyman - Fisher T(X ) =

X(n) .

, , T =T(X

) =X(n) .

X(n) ,

fX(n) (t) = n(FX(t))n1fX(t)

fX() FX() Xi, i = 1, 2, . . . , n ,

fX(t) =1

, t(0, )

FX(t) =

0 , t 0t

, t(0, )

1 , t

,

fX(n) (t) = n

t

n1 1

, t(0, ).

: RR > 0,E((T )) = 0

(t)n

tn1dt = 0 n

(t)tn1dt = 0

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. . .

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.

108 240

(( ))

0 ( ) n n 0 ( ) 0

(t)tn1dt = 0 dd

0

(t)tn1dt = 0()n1 = 0, > 0() = 0, > 0(t) = 0,t > 0, T =T(X

) = X(n) .

2.2.7

: ,

f(x

; ) =

ni=1

f(xi; ) =

ni=1

e(xi)I[,)(xi) =exp

n

i=1

xi + n

n

i=1

I[,)(xi).

X(1) =min{X1, X2, . . . , X n}, ,n

I (x )1 , xi, i = 1, 2, . . . , n 1 , x(1)

I (x )

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.

109 240

i=1

I[,)

(xi) =

0 , .=

0 , .=I

[,)(x

(1

))

,f(x

; ) = exp

n

i=1

xi + n

I[,)(x(1))., h(x

) = exp

n

i=1

xi

q(T(x ), ) = enI[,)(x(1)), Neyman - Fisher T(X

) =X(1)

.

, , T =T(X ) =X(1) .

X(1) ,

fX(1) (t) =n(1 FX(t))n1fX(t)

fX() FX() Xi, i = 1, 2, . . . , n ,

fX(t) = e(t), t[, )

FX(t) =

0 , t < 1 e(t) , t[, ),

fX(1) (t) = n

1 (1 e(t))n1

e(t), t[, ).

: RR > 0,E((T)) = 0

(t)nen(t)dt = 0nen

(t)entdt = 0

(t)entdt =

0 dd

(t)entdt = 0 ()en = 0,R() = 0, R (t) = 0, t R, T =T(X) =X(1) .

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.

110 240

( ) ,

( ) ,

, (

) (1)

2.2.8

: X U(, ), Y =|X| U(0, )., , FY(y) fY(y) =

d

dyFY(y) , ,

.. Y, y > 0

F|X|(y) = P(X y) = P( y X y) = P(X y) P(X y) = FX(y) FX( y)

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.

111 240

| | f|X|(y) = ddy

F|X|(y) = ddy

FX(y) dd(y) FX(y) d(y)d(y) =fX(y) +fX(y) =

1

2+ 1

2 = 1

,

0< y < , Y =|X| U(0, )., 2.2.7,

|X|(n) =max{|X1|, |X2|, . . . , |Xn|}

.

2.2.9

: (. -

2.1.10(1)) ,

, T(X) =X .

T(X) = X ,

.

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. . .

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.

112 240

E(X) = 0, {1, 0, 1} (x) = 0 x .,

E(X) = 03

x=1

(x)P(X =x) = 0

=1 , (1)P1(X = 1) + (2)P1(X = 2) + (3)P1(X = 3) = 0 = 0 , (1)P0(X = 1) + (2)P0(X = 2) + (3)P0(X = 3) = 0

= 1 , (1)P1(X = 1) + (2)P1(X = 2) + (3)P1(X = 3) = 0

(1) 16

+ (2) 16

+ (3) 23

= 0

(1)1

3+ (2)

1

3+ (3)

1

3= 0

(1)2

3+ (2)

1

6+ (3)

1

6= 0

(1) =(2) = (3) = 0.

(x) = 0,x ,

T(X)=

X .

2.2.10

: S , E(S) = 0(s) = 0 S .

(S) =1(S) 2(S) =T1 T2 T1 T2 ,

E(S) = E(1(S) 2(S)) = E1(S) E2(S) = = 0. S (s) = 0

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. . .

Bayes . . .

.

113 240

( )

s S, T1 =T2.

2.2.11

: T (T) =(X1, X2) =X2

1X2 1,

, ,

E(T) = E(X2

1 X2 1) = EX21 EX2 1 = VarX1 + (EX1)2 EX1 1 =

1 + 2 2 1 = 0., 2.1.15, T(X

) = (X1, X2)

.

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. . .

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.

114 240

2.2.12

: ,

f(x

; ) =

ni=1

f(xi; ) =

ni=1

12

e(xi)2

22 =1

(2)n/2n exp

122n

i=1

(xi )2 =exp

n2 ln 2 nln 122n

i=1

x2in

2+

1

q(T(x

); ) = exp

nln 1

22

n

i=1

x2i +1

n

i=1

xi

T(x

) =

n

i=1

xi,

n

i=1

x2i

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. . .

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.

115 240

h(x

) = exp

n

2ln 2n

2

, f(x

; ) = q(T(x

); )h(x

), -

Neyman-Fisher, T(X

) =

ni=1

Xi,

ni=1

X2i

. ,

(T) =1

n(n+ 1)

n

i=1Xi

2

12n

n

i=1X2i

,

E(T) = E

1n(n+ 1) n

i=1

Xi

2

12n

ni=1

X2i

=1

n(n+ 1)

Var n

i=1

Xi

+E

ni=1

Xi

2 12nE

ni=1

X2i

=1

n(n+ 1)(n2 + n22) 1

2n2n2 =2 2 = 0

, 2.1.15, T(X ) . 6.3.2

, .

2.2.13

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. . .

Bayes . . .

.

116 240

:

1. 2.2.3, k = 3 ni = 1, i = 1, 2, 3, B(1, ) , 1,

( )= 3

l (1

) ( ) = 0

( )=

l

( )=

3

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. . .

Bayes . . .

.

117 240

A() ln( ), B(x ) , c() ln 1 D(x ) i=1 xi, 2.1.20, D(x

) =

3i=1

Xi

c() = ln

1 R

R, D(x

) =

3

i=1 Xi .2. ES1 = 1 P(S1 = 1) + 0 P(S1 = 0) = P(S1 = 1) = P(X1 = 1) = ,(0, 1),

S1 .

,

ES1 = 1 P(S1 = 1) + 0 P(S1 = 0) =P(S1 = 1) = P(X1 = 1, X2 = 1) =P(X1 = 1)P(X2 = 1) = =

2,(0, 1)., S2

2.

3. S1 D =

D(x

) =

3i=1

Xi , , 2.1.16

S1

=E(S1|D) .,S

1 =E(S1|D) = 1 P(S1 = 1|D) + 0 P(S1 = 0|D) = P(S1 = 1|D).

,

P(S1 = 1|D =d) = P(X1 = 1, X1 + X2 + X3 =d)P(X1 + X2 + X3 =d)

P(S1 = 1|D =d) = P(X1 = 1, X2 + X3 =d 1)P(X1 + X2 + X3 =d)

. (6.5)

Xi B(1 )

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. . .

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.

118 240

Xi B(1, ), ,D =X1 + X2 + X3 B(3, )fD(d) =

3

d

d(1 )3d (6.6)

X1 + X2 B(2, )fX1+X2 (d 1) =

2

d 1

d1(1 )2(d1) (6.7)

, (6.5), (6.6) (6.7), .

P(S1 = 1|D =d) = 2

d 1 d1(1 )2(d1) 3

d

d(1 )3d

= d

3.

,P(S1 = 1|D) =E(S1|D) = D3

= 1

3

ni=1

Xi = X .

, S2

= E(S2 = 1|D) = P(S2 = 1|D) = D(D 1)6

2.

, g() = P(X1 X2) = 1P(X1 < X2) = 1P(X1 = 0, X2 = 1) =1 (1 ) = 1 + 2, 1 D

3+

D(D 1)6

g().

2.2.14

: 2.1.20,

ni=1

Xi,

ni=1

X2i

., 1.1.11,

S2

=1

1

n(Xi X )

2=

1

1 n

X2

i

1 nXi2

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. . .

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.

119 240

S n 1 i=1

(Xi X) n 1 i=1

Xi n i=1

Xi

, 2

, ,

2.1.17, S2 2.

k Sk

, ,

k.

2.1.22,

T =(n 1) S2

2 X2n1.

ESk =E(S2)k/2 =E

n 1

22

n 1 S2

k/2=

k

(n 1)k/2 ETk/2, T X2n1.

,

ETk/2 =

0

tk/2 1

((n 1)/2)2(n1)/2 t(n1)/21et/2dt =

1

((n 1)/2)2(n1)/2

0

t(k+n1)/21et/2dt =

1

((n 1)/2)2(n1)/2 2(k+n1)/2

((n 1 + k)/2) = 2k/2 ((n 1 + k)/2)((n 1)/2) .

n 1

2

k/2((n 1)/2)

((n 1 + k)/2) Sk

k.

2.2.15

5/19/2018 1-

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. . .

Bayes . . .

.

120 240

:

1. 2.2.7 T(X

) =X(n) =max{X1, X2, . . . , X n} , ,

fT(t) =n

ntn1, t(0, ).

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. . .

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.

121 240

T

,

ET =

0

t n

ntn1dt =

n

n+ 1E

n+ 1

n X(n)

= , > 0.

, 2.1.17, n+ 1

n X(n)

.

r

, r > 0 , .

T = X(n)

, Tr

r.

,

ETr =

0

tr nn

tn1dt = nn+ r

r E

n+ rn

Xr(n)

= r, > 0.

, n+ r

n Xr

(n)

r, r > 0.

S(T) r,

T, S(T),

2.1.17, r, ,

ES(T) = r

0

s(t)n

ntn1dt =r n

0

s(t)tn1dt =n+r n dd

0

s(t)tn1dt =

d

d n+r

ns()n

1

= (n + r)n+r

1

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d ns() = (n+ r) s() =

n+ r

n r s(t) = n+ r

n tr.

, n+ r

n Xr

(n)

r, r > 0.

2. Sc = cX(n) -

... c.

...(Sc, ) = VarSc + (ESc )2 (6.8)

,

ESc =E(cX(n)) = cEX(n) =c n

n+ 1.

Var

Sc

=Var(

cX(n))

= c2VarX

(n) =c2 E

X2

(n)+

(E

X

(n))

2 = c2 nn+ 2

2

n2

(n+ 1)22 =

c2 n

(n+ 2)(n+ 1)22.

, 6.8 ,

...(Sc, ) = c2

n

(n+ 2)(n+ 1)22 + (c

n

n+ 1 )2

...(Sc, ) =

n

(n+ 2)(n+ 1)2 +

n2

(n+ 1)2

2c2 2n

n+ 12c+ 2.

,

c = 2n

n+ 12

2 n

( 2)( 1)2 +n2

( 1)2

2

=n+ 2

n+ 1.

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2

(n+ 2)(n+ 1)2 + (n+ 1)2

, , ..., S = n+ 2

n+ 1X(n).

6.3.3 ,n+ 1

n X(n)

cX(n) S =n+ 2

n+ 1X(n),

... (. 1.1.8).

2.2.16

: .

1 (), ,

1. S ={x

:f(x

; )> 0} = Rn+ .

2. f(x

, ) =

n

k=1

1

(k)kxk1k e

xk =

1

(n

k=1 (k))n(n+1

)/2

n

k=1

xk1k e

xk

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(k) (

(k))

= exp

n

k=1

ln (k) n(n+ 1)2

ln +

nk=1

(k 1) ln xk 1

nk=1

xk

. ,

A() =n

k=1

ln (k) n(n+ 1)2

ln , B(x

) =

nk=1

(k 1) ln xk, c() = 1

D(x

) =

nk=1

xk .

2.1.20, D(X

) =

nk=1

Xk -

c() = 1 R R,

.

, -

D(X

) =

nk=1

Xk. ED(X

) ,

ED(X

) =E

nk=1

Xk =

nk=1

EXk =

nk=1

k = n(n+ 1)

2

E

2n(n+ 1)n

k=1

Xk

=.

2

n(n+ 1)

nk=1

Xk, -

D(X

),

(. 2.1.17). n

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( )

D =

nk=1

Xk

, Dr r. -

Xk Gamma(k, ), k = 1, 2, . . . , n , ..

D =

n

k=1XkGamma(

n

k=1k, )Gamma( n(n+ 1)

2, ).

EDr

=

0

xr 1

(n(n+ 1)/2)n(n+1)/2xn(n+1)/21e

x dx =

(n(n+ 1)/2 + r)

(n(n+ 1)/2)r

E

(n(n+ 1)/2)

(n(n+ 1)/2 + r)Dr

= r, > 0 r >n(n+ 1)2

.

, 2.1.17, (n(n+ 1)/2)

(n(n+ 1)/2 + r)Dr

r.

2.2.17

: (. -

2.1.10(1)) ,

, T(X) =X .

T(X) = X ,

.

E(X) = 0,

{0, 1

} (x) = 0

x

.

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{ },

E(X) = 02

x=1

(x)P(X =x) = 0

= 0 , (1)P0(X =1) + (2)P0(X = 2) = 0 = 1 , (1)P1(X =1) + (2)P1(X = 2) = 0

(1) 23

+ (2)1

3= 0

(1) 13

+ (2)2

3= 0

(1) = (2) = 0. (x) = 0,x , T(X) = X .

, ,

= 0, E0X = (1)P0(X =1) + 2P0(X = 2) =23

+ 21

3= 0

= 1, E1X = (1)P1(X =1) + 2P1(X = 2) =1

3 + 2

2

3 = 1

,

={0, 1}, EX =, X

,

.

2.2.18

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: .

1 (), ,

1. S={x

:f(x

; )> 0} ={0, 1, . . .} {0, 1, . . .} . . . {0, 1, . . .} .

2. f(x

, ) =

n

i=1

eti(ti)

xi

xi!=e

ni=1ti

ni=1xi

n

i=1

txii

n

i=1

1

xi!

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= exp

n

i=1

ti +

ni=1

xiln tin

i=1

ln xi! + ln

ni=1

xi

.,

A() =n

i=1

ti, B(x

) =

ni=1

xiln tin

i=1

ln xi!, c() = ln D(x

) =

ni=1

xi

.

2.1.20, D(X

) =

ni=1

Xi -

c() = ln R R, .

XiP(ti),

D =

D(X ) =

ni=1 X

i P(n

i=1 ti),

ED =

ni=1

ti E

1ni=1ti

D

= , > 0. (6.9)

2.1.17 1n

i=1tiD

, D

.

,

ED2

=VarD+ (ED)2

=

n

i=1

ti + (

n

i=1

ti)22.

6 9

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, 6.9,

ED2

=ED+ (

ni=1

ti)22 E

D(D 1)

(

ni=1

ti)2

=2, > 0.

, , D(D 1)

(

ni=1

ti)2

.

2.2.19

: X

2(. 1.1.10),

EX =

2E(2 X) = , > 0.

2 X . 2.2.7

X(n) =max{X1, X2, . . . , X n

} , -

, 2.1.16, S =E(2 X|X(n)) 1

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. 2.2.16 n+ 1

n X(n) , ,

,

, , E(2 X|X(n)) = n+ 1

n X(n)E(

2n

n+ 1X|X(n)) =

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