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11.1
. . : . , . , . n , .
, . , . , .
1907 , .. . , . .
: S = { , , , }. . . si sj pij.
pij . , pii. S, . . .. . .
11.1 , , . . , . , . , . . R,N S. .
11.1 . , . .
i , . pij(2). 11.1 : 1) 2) 3) . , . p11p13. , .
P(2)13 = p11p13 + p12p23 + p13p33 .
, o . 1,3 P . ,
.
11.1 . ij pij(n) Pn si sj n .
. ( 17).
11.2 ( 11.1 ) . . . MatrixPowers . MatrixPowers 1-6. 11.1.
, . R,N S, 0,4 0,2 0,4 . . . , .
, . r 1. it- U si. , . 11.2 , u . si n i-
. ( 18.)
si, u i- 1 0.
11.3 ( 11.1) (1/3;1/3;1/3;). 11.2 P3.
)
.
11.4 . , , . b . .
11.5 , , . .
11.6 , , . 80% , 40% , 70% ,20% 10% .
11.7 11.6 .
11.8 ( ) , . 11.5. . , . .
11.9 ( ) , , G g. GG Gg( gG) gg. GG Gg G g. GG , gg Gg. , , . O . , . G G g . . G g . GG, Gg gg.
. . . . . , GG,Gg gg .
11.10 11.9 : , .
11.11 , , , . . . : s1 = (GG; GG), s2 = (GG; Gg), s3 = (GG; gg), s4 = (Gg; Gg), s5 = (Gg; gg), s6 = (gg; gg).
s2. , . , . , s1 , s2 s4 . . :
11.12 ( ) . . -- , . . , S , S S . . , . . kn2 , . .
11.1: .
11.2: 10.000 .
. 11.2. 1 , . . .
1. . . w (1) , w(2) w(3) P, P2 P3.
2. 11,4, =0 -1/2 . P,P2 P3. Pn? Pn n ? .
3. 11.5, P,P2 P3. Pn?
4. 11.6 .
5. 11.7, .
6. 11.9, . w(1), w(2) w(3). w(n)?
7. P2, P3, P4 Pn . P= . .
8. 0 1 . . , p q=1-p . 0 1. ?
9. 8, 0 . 0? 0? p=.1. MatrixPowers 100 . . p=.2. 100 ?
10. MatrixPowers, n, Pn , n=1 N. An Pn.
11. , . 80% , 10% 10% . , 60% , 20% , 20% . , 50% 25% . , . .
12. 11, . .8. . 11. . .
13. u(n) u P. u(n) u=(0,1,0), u=(1/3,1/3,1/3).
14. MatrixPowers, P1 P6 11.9 11.10. .
15. P. .
16. 15 n . 11.1 11.8. n ?
17. 11.1.
18. 11.2.
19. . . . . . .
a) n- , (n+1) ?
b) . (), .
c) (n-1) n- .
d) ?