Теория вероятностей и математическая статистика. Билялов Р.Ф. - 108 стр.

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P (ξ
0
= 1, ξ
1
= 3, ξ
2
= 3, ξ
3
= 2) = .7 · .4 · .3 · .4 =
336
10000
.
½
(q
1
, q
2
, q
3
)P = (q
1
, q
2
, q
3
)
q
1
+ q
2
+ q
3
= 1
→
q
1
+ 6q
2
+ 3q
3
= 10q
1
5q
1
+ 2q
2
+ 4q
3
= 10q
2
q
1
+ q
2
+ q
3
= 1
→
q
1
= 16/47
q
2
= 17/47
q
3
= 14/47
.
ξ
t
, t = 1, 2, ...,
P (ξ
t
= 1) = p, P (ξ
t
= −1) =
1 − p. η
t
= ξ
t
ξ
t+1
;
η
t
= ξ
1
ξ
2
···ξ
t
; η
t
= ϕ(ξ
t
, ξ
t+1
), ϕ(−1, −1) = 1, ϕ(−1, 1) = 2,
ϕ(1, −1) = 3, ϕ(1, 1) = 4?
P (η
t+1
= 1|η
t−1
= η
t
= 1) =
P (η
t−1
= η
t
= η
t+1
= 1)
P (η
t−1
= η
t
= 1)
=
=
P (ξ
t−1
= ξ
t
= ξ
t+1
= ξ
t+2
= ±1)
P (ξ
t−1
= ξ
t
= ξ
t+1
= ±1)
=
p
4
+ q
4
p
2
+ q
2
,
P (η
t+1
= 1|η
t
= 1) =
p
3
+ q
3
p
2
+ q
2
.
p = q, ²
1
, ²
2
, ²
3
, ±1,
P (η
t+1
= ²
1
|η
t−1
= ²
2
, η
t
= ²
3
) = P (η
t+1
= ²
1
|η
t
= ²
3
).
η
t
P (η
t+1
= 1|η
t
= 1) = 0.5, P (η
t+1
= −1|η
t
= −1) =
p
2
q + pq
2
2pq
= 0.5,
                                                                336
   2) P (ξ0 = 1, ξ1 = 3, ξ2 = 3, ξ3 = 2) = .7 · .4 · .3 · .4 = 10000 .
   3)
                                            
    ½                                        q1 + 6q2 + 3q3 = 10q1
       (q1 , q2 , q3 )P = (q1 , q2 , q3 )
                                          →   5q1 + 2q2 + 4q3 = 10q2 →
       q1 + q2 + q3 = 1                     
                                              q1 + q2 + q3 = 1
                              
                               q1 = 16/47
                                q2 = 17/47 .
                              
                                q3 = 14/47
   Çàäà÷à 8.4. Ïóñòü ξt , t = 1, 2, ...,  íåçàâèñèìûå îäèíàêîâî ðàñ-
ïðåäåëåííûå ñëó÷àéíûå âåëè÷èíû. P (ξt = 1) = p, P (ξt = −1) =
1 − p. ßâëÿþòñÿ ëè öåïüþ Ìàðêîâà âåëè÷èíû: à) ηt = ξt ξt+1 ; á)
ηt = ξ1 ξ2 · · · ξt ; â) ηt = ϕ(ξt , ξt+1 ), ãäå ϕ(−1, −1) = 1, ϕ(−1, 1) = 2,
ϕ(1, −1) = 3, ϕ(1, 1) = 4? Äëÿ öåïåé Ìàðêîâà íàéòè ìàòðèöû âåðî-
ÿòíîñòåé ïåðåõîäà çà îäèí øàã.
   Ðåøåíèå.
   à) Ðàññìîòðèì äâå ÷àñòíûå óñëîâíûå âåðîÿòíîñòè.

                                         P (ηt−1 = ηt = ηt+1 = 1)
        P (ηt+1 = 1|ηt−1 = ηt = 1) =                              =
                                             P (ηt−1 = ηt = 1)

                 P (ξt−1 = ξt = ξt+1 = ξt+2 = ±1)  p4 + q 4
             =                                    = 2       ,
                     P (ξt−1 = ξt = ξt+1 = ±1)     p + q2
                                                p3 + q 3
                      P (ηt+1 = 1|ηt = 1) =              .
                                                p2 + q 2
Èç ðàâåíñòâà äâóõ óñëîâíûõ âåðîÿòíîñòåé ñëåäóåò, ÷òî p=q=0.5. Íî
åñëè p = q, òî äëÿ ëþáûõ ²1 , ²2 , ²3 , ðàâíûõ ±1, èìååì:

         P (ηt+1 = ²1 |ηt−1 = ²2 , ηt = ²3 ) = P (ηt+1 = ²1 |ηt = ²3 ).

   ò.å. ηt îáðàçóþò öåïü Ìàðêîâà.

                                                             p2 q + pq 2
  P (ηt+1 = 1|ηt = 1) = 0.5, P (ηt+1 = −1|ηt = −1) =                     = 0.5,
                                                                 2pq


                                      108

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