Теория вероятностей и математическая статистика. Билялов Р.Ф. - 116 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

k = 1, 2, ..., t = 0, 1, 2, ... ξ
k
(t)
k,
t.
µ
0
= 1,
µ
1
= ξ
t
(0).
µ
2
=
½
0, µ
1
= 0,
ξ
1
(1) + ξ
2
(1) + ··· + ξ
µ
1
(1), µ
1
6= 0
µ
t+1
=
½
0, µ
t
= 0,
ξ
1
(t) + ξ
2
(t) + ··· + ξ
µ
t
(t), µ
t
6= 0
.
A
t
= Mµ
t
− t.
Φ
t
= Mx
µ
t
ϕ(x) = Mx
ξ
k
(t)
=
∞
P
m=0
x
m
p
m
, p
m
= P (ξ
k
(t) = m). a = ϕ
0
(x)|
x=1
=
P
mp
m
=
Mξ
k
(t). Φ
1
= M ξ
µ
1
= M x
ξ
1
(0)
= ϕ(x),
Φ
t+1
= Mx
µ
t+1
=
∞
X
m=0
x
m
P (µ
t+1
= m) =
=
∞
X
m=0
x
m
∞
X
k=0
P (µ
t
= k)P (ξ
1
+ ··· + ξ
k
= m) =
=
∞
X
k=0
P (µ
t
= k)
∞
X
m=0
x
m
P (ξ
1
+ ··· + ξ
k
= m) =
∞
X
k=0
P (µ
t
= k)·
·Mx
ξ
1
+···+ξ
k
==
∞
X
k=0
P (µ
t
= k)ϕ(x)
k
= Φ
t
(ϕ(x)),
Φ
t+1
, Φ
t
ϕ
A
t
= Mµ
t
= (Φ
t
)
0
x
|
x=1
A
t+1
= (Φ
t+1
)
0
x
|
x=1
(Φ
t
(ϕ(x)))
0
x
|
x=1
=
dΦ
t
(ϕ)
dϕ
ϕ
0
x
|
x=1
= A
t
a,
A
t
= a
t
. t → ∞
a < 1, A
t
→ 0
a = 1, A
t
= 1
a > 1, A
t
→ ∞
k = 1, 2, ..., t = 0, 1, 2, ... Ñëó÷àéíàÿ âåëè÷èíà ξk (t) èíòåðïðåòèðóåòñÿ
êàê ñëó÷àéíîå ÷èñëî ïîòîìêîâ ÷àñòèöû ñ íîìåðîì k, ñóùåñòâîâàâ-
øåé â ìîìåíò t. Ïóñòü â íà÷àëüíûé ìîìåíò èìååòñÿ îäíà ÷àñòèöà:
µ0 = 1, òîãäà ÷èñëî ÷àñòèö â ñëåäóþùèé ìîìåíò âðåìåíè ðàâíî:
µ1 = ξt (0). Äëÿ ïîñëåäóþùèõ ìîìåíòîâ èìååì
                      ½
                          0, µ1 = 0,
                µ2 =
                          ξ1 (1) + ξ2 (1) + · · · + ξµ1 (1), µ1 6= 0
è                        ½
                             0, µt = 0,
                µt+1 =                                                  .
                             ξ1 (t) + ξ2 (t) + · · · + ξµt (t), µt 6= 0
Âû÷èñëèì At = M µt − ñðåäíåå ÷èñëî ÷àñòèö â ìîìåíò t. Äëÿ ýòîãî
ðàññìîòðèì ïðîèçâîäÿùèå ôóíêöèè Φt = M xµt è ϕ(x) = M xξk (t) =
 P
 ∞                                                                        P
    xm pm , ãäå pm = P (ξk (t) = m). Ïîëîæèì a = ϕ0 (x)|x=1 = mpm =
m=0
M ξk (t). Î÷åâèäíî, ÷òî Φ1 = M ξ µ1 = M xξ1 (0) = ϕ(x), äàëåå ðàññìàò-
ðèâàåì
                                 X∞
               Φt+1 = M xµt+1 =      xm P (µt+1 = m) =
                                             m=0
                     ∞
                     X          ∞
                                X
                 =         xm       P (µt = k)P (ξ1 + · · · + ξk = m) =
                     m=0     k=0
          ∞
          X                  ∞
                             X                                    ∞
                                                                  X
      =         P (µt = k)         xm P (ξ1 + · · · + ξk = m) =         P (µt = k)·
          k=0                m=0                                  k=0
                                      ∞
                                      X
                ·M xξ1 +···+ξk ==           P (µt = k)ϕ(x)k = Φt (ϕ(x)),
                                      k=0
Íàéäåííóþ ñâÿçü ìåæäó Φt+1 , Φt è ϕ èñïîëüçóåì äëÿ âû÷èñëåíèÿ
At = M µt = (Φt )0x |x=1 ñëåäóþùèì îáðàçîì:
                                                        dΦt (ϕ) 0
      At+1 = (Φt+1 )0x |x=1 (Φt (ϕ(x)))0x |x=1 =               ϕx |x=1 = At a,
                                                          dϕ
îòêóäà At = at . Â ïðåäåëå t → ∞ íàõîäèì:
   a < 1, At → 0  äîêðèòè÷åñêèé ïðîöåññ;
   a = 1, At = 1  êðèòè÷åñêèé ïðîöåññ;
   a > 1, At → ∞  íàäêðèòè÷åñêèé ïðîöåññ.

                                              116

Страницы