- pi 2014

1. (25) 3 pi pi i pi, i = 1, 2, 3, . 1 . 2 pi . 2 3 1 .() pi pi pi- pi pi .() .() pi 1 , pi pi ;

pi() pi 0 < pi < 1, i = 1, 2, 3. n = 1, 2, Xn pi n- , Xn {1, 2, 3}. Xn = i, i pi pi pi . pi , pi pi pi pi pi pi pi . pi , pi. pi Xn, n = 1, 2, . . . pi pi pi pi:

P =

1 p1 p1 00 1 p2 p2p3 0 1 p3

3

21

1 p1 p1 1 p2

p2

1 p3

p3

1: 1:

() pi pipi, pipi. pipi i, pii > 0, pi. pi pi pi =(pi1, pi2, pi3) pi :

pi1 = (1 p1)pi1 + p3pi3pi2 = (1 p2)pi2 + p1pi1

pi1 + pi2 + pi3 = 1

1

pi pi pipi

pi1p1 = pi2p2 = pi3p3

pi1 + pi2 + pi3 = 1.

pi

pi1 =

1p1

1p1

+ 1p2 +1p3

, pi2 =

1p2

1p1

+ 1p2 +1p3

, pi3 =

1p3

1p1

+ 1p2 +1p3

.

() 1 pi ( ) 0 pi , Xn = i,

Rn(Xn, Xn+1) =

{1, pi pi0, pi 1 pi

pi pi i

r(i) = E(Rn(Xn, Xn+1)|Xn = i) = pi. pi pi pi

C =iS

r(i)pii =3i=1

pi

1pi

1p1

+ 1p2 +1p3

=3

1p1

+ 1p2 +1p3

.

2. (30) , pi pi. pi pi p pi 1 p. pi pi pi pi . pi pi pi .() pi pi;() pi pi ;() pi pi pi pi pi pi;

pi: {Xn, n = 0, 1, . . .} pi X0 = 0 Xn n- pi.

pi() Xn n- pi. Xn = 2 2, pi , . Xn = 1, 0, 1 pi pi 1 ( pi 1 p) 1 (pi p). pi, pi , Xn, Xn+1 pi . pi

-2 -1 0 1 21

p

1 p

p

1 p

p

1 p1

2

-2 2 pi -1, 0, 1 pi. - pi pi 0 pi -2 2. pi, fi pi i pi -2 2. f1, f0, f1 pi pi , pi pipi pi pi :

f1 = 1 + pf0f0 = 1 + (1 p)f1 + pf1f1 = 1 + (1 p)f0.

pi pi:

f0 = 1 + (1 p) + (1 p)pf0 + p+ p(1 p)f0 = 2 + 2p(1 p)f0 pi

f0 =2

1 2p(1 p)pi pi pi.() pi pi pi pi 2 0. xi pi pi 2 X0 = i. x2 = 0, x2 = 1. x1, x0, x1 pi pi , pi pipipi pi :

x1 = px0x0 = (1 p)x1 + px1x1 = p 1 + (1 p)x0.

pi pi:

x0 = (1 p)px0 + p2 + p(1 p)x0 = p2 + 2p(1 p)x0 pi

x0 =p2

1 2p(1 p)pi pi pi .() pi pi p pi.pi pi pipi pi x0 p. x0 pi:

p2

1 2p(1 p) pp

1 2p(1 p) 1 p 1 2p(1 p) 1 3p+ 2p2 0.

2p2 3p+ 1 1 12 . pi 12 p 1, pi pi pi ( pi pi ).

3. (25) {Xn, n = 0, 1, . . .}

3

S = {1, 2, . . . , 7} pi pi pi

P =

14

34 0 0 0 0 0

1 0 0 0 0 0 015

15

15

15

15 0 0

0 1414 0 0

14

14

0 0 0 0 1 0 00 0 0 0 0 0 10 0 0 0 0 1 0

.

() pi pi pipi pi.() pi pi :

limn p

(n)12 , limn p

(n)32 , limn p

(n)17 , limn p

(n)45 , limn p

(n)42 , limn p

(n)34 .

() X0 = 6, pi pi pi pi pi 7;() X0 = 1, pi pi pi pi pi 1;

pi() pi pi pi pi pi:

1

2

3

4

6

5

7

14

34 1

151

5

15

15

1515

14

14

14

14

1

1 1

pi pi pi:

1. C1 = {1, 2} pipi pipi 2. C2 = {5} pi 3. C3 = {6, 7} pipi pipi 4. C = {3, 4} pi pi

4

pi pipi, C1 pi pi p11 > 0 C3 pi pi 2.() pipi .

pi C2 pi pi pi pi

pi1 =1

4pi1 + pi2

pi1 + pi2 = 1

pi pi pipi pi1 = 47 , pi2 =37 .

C2 pi pi pi pi 5 pi pi pi5 = 1.

C3 pi pi pi, pi pi pi :

pi6 = pi7

pi6 + pi7 = 1

pi pi pipi pi6 = pi7 = 12 . pi pipi pi pi pi pi

pi pi pi pipi . C1 pi pi x3,C1 , x4,C1 pi -

x3,C1 = p31 1 + p32 1 + p33x3,C1 + p34x4,C1 + p35 0x4,C1 = p42 1 + p43x3,C1 + p46 0 + p47 0

pi

x3,C1 =2

5+

1

5x3,C1 +

1

5x4,C1

x4,C1 =1

4+

1

4x3,C1

pi pi pipi x3,C1 =35 , x4,C1 =

25 .

C2 pi pi x3,C2 , x4,C2 pi -

x3,C2 = p34 1 + p33x3,C2 + p34x4,C2x4,C2 = p43x3,C2

pi

x3,C2 =1

5+

1

5x3,C2 +

1

5x4,C2

x4,C1 =1

4x3,C2

pi pi pipi x3,C2 =415 , x4,C2 =

115 .

C3 pi pi pipi pi :

x3,C3 = 1 x3,C1 x3,C2 = 13

5 4

15=

2

15

x4,C3 = 1 x4,C1 x4,C2 = 12

5 1

15=

8

15.

5

pi pi.

pi limn p(n)12 = pi1 =

47 , C1 pipi pi

pipi pi.

pi limn p(n)32 = x3,C1pi2 =

3537 =

935 , pi pipi pi C1

pi pi 2.

pi limn p(n)17 = 0 1 pi

pi C1. pi :

limn p

(n)45 = x4,C2pi5 =

1

15

limn p

(n)42 = x4,C1pi2 =

1

15

limn p

(n)34 = 0,

pi 4 pi.() 6, pi pi pipi C3. pi , pi pi6, pi7 pi pi pi 6 7 , pi pi pi7 = 12 .() 1, pi pi pipi C1, pi pi pi1 = 47 .

4. (30) pi pi pi pi Poisson(). pi pi pi - . X(t) pi t.() {X(t), t 0} .() pi pi Chapman-Kolmogorov pi - pij(t), i, j S.() pi pipi , . ;() 100 pi pi - 1 , pi pi pi pi ; (pi pi pi .)

pi () pi pi pi . pi X(t), t 0 -.

n = , n = 0, 1, . . .. pi n pi

pi pi pi n pi. pi pi - n, n = n, n = 0, 1, 2, . . ..

6

0 1 2 n 1 n

() pi Chapman-Kolmogorov :

pi0(t) = pi0(t) + pi1(t), i 0,pij(t) = (+ j)pij(t) + pi,j1(t) + (j + 1)pi,j+1(t), i 0, j > 0

() . pi pi, pi pi, pipi pipi. , pipipipi pi . pipi pi. pi pi {0, 1, . . . , n}, {n+ 1, . . .}, n = 0, 1, . . ., pi

p0 = p1

p1 = p22

pn1 = pnn

pi pi pipi

p1 =

p0, p2 =

2p1 =

2

22p0, . . . , pn =

n

n!np0

pi pn pi pin=0

pn = p0

n=0

n

n!n= 1.

n=0

n

n!n=n=0

(

)nn!

= e

pipi , > 0, pi pi pi pipi

p0 = e

pn = e

(

)nn!

> 0, n = 0, 1, 2, . . .

pi pipi pi. , pi Poisson pi .() pi () pi X(t) t pi Poisson pi . pi

limtE(X(t)) =

.

, = 100 60 = 6000, 1/ = 1, pilimtE(X(t)) = 6000.

7