Теория вероятностей и математическая статистика. Билялов Р.Ф. - 109 стр.

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P =
·
0.5 0.5
0.5 0.5
¸
.
P (η
t+2
= k|η
t
= i, η
t+1
= j) =
= P (η
t
= i, η
t+1
= j, η
t+2
= k)/P (η
t
= i, η
t+1
= j) =
P (η
t
= i, ξ
t+1
= j/i, ξ
t+2
= k/j)/P (η
t
= i, ξ
t+1
= j/i) =
= P (ξ
t+2
= k/j) = P (η
t+2
= k|η
t+1
= j).
ϕ − ϕ
−1
−
ϕ
−1
= (ϕ
−1
1
, ϕ
−1
2
) :
P (η
t+2
= k|η
t
= i, η
t+1
= j) =
= P (η
t
= i, η
t+1
= j, η
t+2
= k)/P (η
t
= i, η
t+1
= j) =
= P (ξ
t
= ϕ
−1
1
(i), ξ
t+1
= ϕ
−1
2
(i), ξ
t+1
= ϕ
−1
1
(j), ξ
t+2
= ϕ
−1
2
(j),
ξ
t+2
= ϕ
−1
1
(k), ξ
t+3
= ϕ
−1
2
(k))/
P (ξ
t
= ϕ
−1
1
(i), ξ
t+1
= ϕ
−1
2
(i), ξ
t+1
= ϕ
−1
1
(j), ξ
t+2
= ϕ
−1
2
(j)) =
= δ
ϕ
−1
2
(j),ϕ
−1
1
(j)
δ
ϕ
−1
2
(j),ϕ
−1
1
(k)
P (ξ
t+3
= ϕ
−1
2
(k)) = P (η
t+2
= k|η
t+1
= j),
δ
m,n
−
p
11
= P (η
t+1
= 1/η
t
= 1) = δ
ϕ
−1
2
(1),ϕ
−1
1
(1)
P (ξ
t+1
= ϕ
−1
1
(1)) = q,
P =
q p 0 0
0 0 q p
q p 0 0
0 0 q p
.
N
µ
0
(n)
n
µ
0
(n), n = 0, 1, 2, ...
                                      ·                ¸
                                          0.5 0.5
                                P =                        .
                                          0.5 0.5
á)
                        P (ηt+2 = k|ηt = i, ηt+1 = j) =
           = P (ηt = i, ηt+1 = j, ηt+2 = k)/P (ηt = i, ηt+1 = j) =
          P (ηt = i, ξt+1 = j/i, ξt+2 = k/j)/P (ηt = i, ξt+1 = j/i) =
                 = P (ξt+2 = k/j) = P (ηt+2 = k|ηt+1 = j).
Öåïü Ìàðêîâà.
   â) Òàê êàê ϕ − ôóíêöèÿ äâóõ ïåðåìåííûõ, òî ϕ−1 − äâóõêîìïî-
íåíòíàÿ ôóíêöèÿ: ϕ−1 = (ϕ−1   −1
                         1 , ϕ2 ) :


                        P (ηt+2 = k|ηt = i, ηt+1 = j) =
           = P (ηt = i, ηt+1 = j, ηt+2 = k)/P (ηt = i, ηt+1 = j) =
      = P (ξt = ϕ−1             −1             −1             −1
                 1 (i), ξt+1 = ϕ2 (i), ξt+1 = ϕ1 (j), ξt+2 = ϕ2 (j),

                        ξt+2 = ϕ−1             −1
                                1 (k), ξt+3 = ϕ2 (k))/

      P (ξt = ϕ−1             −1             −1             −1
               1 (i), ξt+1 = ϕ2 (i), ξt+1 = ϕ1 (j), ξt+2 = ϕ2 (j)) =

= δϕ−1 (j),ϕ−1 (j) δϕ−1 (j),ϕ−1 (k) P (ξt+3 = ϕ−1
                                               2 (k)) = P (ηt+2 = k|ηt+1 = j),
      2      1      2      1

ãäå δm,n − ñèìâîë Êðîíåêåðà.

p11 = P (ηt+1 = 1/ηt = 1) = δϕ−1 (1),ϕ−1 (1) P (ξt+1 = ϕ1−1 (1)) = q, è ò.ä.
                                      2        1

                                                        
                                    q      p       0   0
                                   0      0       q   p 
                               P =
                                   q
                                                         .
                                           p       0   0 
                                    0      0       q   p
     Çàäà÷à 8.6.  N ÿ÷åéêàõ ïîñëåäîâàòåëüíî íåçàâèñèìî äðóã îò
äðóãà ðàâíîâåðîÿòíî ðàçìåùàþò ÷àñòèöû. Ïóñòü µ0 (n)  ÷èñëî ÿ÷å-
åê, îñòàâøèõñÿ ïóñòûìè ïîñëå ðàçìåùåíèÿ n ÷àñòèö. Ïîêàçàòü, ÷òî
ïîñëåäîâàòåëüíîñòü µ0 (n), n = 0, 1, 2, ... ÿâëÿåòñÿ öåïüþ Ìàðêîâà.
Íàéòè âåðîÿòíîñòè ïåðåõîäà.


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