Теория вероятностей и математическая статистика. Билялов Р.Ф. - 121 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

µ
T − t
1
− (t
2
− t
1
)
T − t
1
1−k
1
−(k
2
−k
1
)
C
k
3
−k
2
l−k
2
µ
T − t
2
T − t
2
k
3
−k
2
0
(l−k
2
)−(k
3
−k
2
)
=
=
(λT )
k
3
k
3
!
e
−λT
· C
k
1
l
µ
t
1
T
k
1
µ
T − t
1
T
k
3
−k
1
· C
k
2
−k
1
k
3
−k
1
µ
t
2
− t
1
T − t
1
k
2
−k
1
·
·
µ
T − t
2
T − t
1
k
3
−k
2
= λ
k
3
e
−λT
t
k
1
1
(t
2
− t
1
)
k
2
−k
1
(T − t
2
)
k
3
−k
2
k
1
!(k
2
− k
1
)!(k
3
− k
2
)!
.
µ
t
−
F (t, x) =
M(x
µ
t
|µ
0
= 1); F (t, x); A(t) = M(µ
t
|µ
0
=
1).
µ
t
− t.
k i (i = 1, .., k)
µ
i
t
, (µ
t
|µ
0
= 1),
M(x
k
i=1
µ
i
t
|µ
1
0
= ··· = µ
k
0
= 1) =
k
Y
i=1
M(x
µ
i
t
|µ
i
0
= 1) = F (t, x)
k
.
F (t + h, x) = M(x
µ
t+h
|µ
0
= 1) =
=
∞
X
k=0
P (µ
h
= k|µ
0
= 1)M(x
µ
t+h
|µ
0
= 1, µ
h
= k) =
=
∞
X
k=0
P (µ
h
= k|µ
0
= 1)M(x
µ
t+h
|µ
h
= k) =
=
∞
X
k=0
P (µ
h
= k|µ
0
= 1)M(x
µ
t
|µ
0
= k) =
∞
X
k=0
P (µ
h
= k|µ
0
= 1)·
·F (t, x)
k
= µh + (1 − (λ + µ)h)F + λhF
2
+ ··· .
dF
dt
= lim
h→0
F (t + h, x) − F (t, x)
h
= µ − (λ + µ)F + λF
2
.
µ                              ¶1−k1 −(k2 −k1 )             µ            ¶k3 −k2
    T − t1 − (t2 − t1 )                            k3 −k2       T − t2
                                                  Cl−k                             0(l−k2 )−(k3 −k2 ) =
         T − t1                                        2        T − t2
                           µ ¶k1 µ         ¶                  µ         ¶k2 −k1
      (λT )k3 −λT       k 1 t1      T − t1 k3 −k1       k2 −k1 t2 − t1
    =          e     · Cl                           · Ck3 −k1                   ·
        k3 !                T         T                         T − t1
         µ          ¶
             T − t2 k3 −k2            tk1 (t2 − t1 )k2 −k1 (T − t2 )k3 −k2
        ·                   = λk3 e−λT 1                                   .
             T − t1                        k1 !(k2 − k1 )!(k3 − k2 )!
     Çàäà÷à 10.5. Ïóñòü µt − âåòâÿùèéñÿ ïðîöåññ ñ íåïðåðûâíûì
âðåìåíåì. Íàéòè: à) äèôôåðåíöèàëüíîå óðàâíåíèå äëÿ F (t, x) =
M (xµt |µ0 = 1); á) ÿâíîå âûðàæåíèå äëÿ F (t, x); â) A(t) = M (µt |µ0 =
1).
    Ðåøåíèå. a) µt − ÷èñëî ÷àñòèö â ìîìåíò t. Ecëè â íà÷àëüíûé
ìîìåíò áûëî k ÷àñòèö, òî êàæäàÿ i àÿ ÷àñòèöà (i = 1, .., k) â ìîìåíò
t äàñò ñëó÷àéíîå ÷èñëî ïîòîêîâ µit , ðàñïðåäåëåííîå êàê (µt |µ0 = 1),
ïîýòîìó
                P
                k
                                                     k
                    µit                              Y             i
       M (xi=1 |µ10 = · · · = µk0 = 1) =                    M (xµt |µi0 = 1) = F (t, x)k .
                                                     i=1

                               F (t + h, x) = M (xµt+h |µ0 = 1) =
                    ∞
                    X
                =           P (µh = k|µ0 = 1)M (xµt+h |µ0 = 1, µh = k) =
                    k=0
                            ∞
                            X
                      =           P (µh = k|µ0 = 1)M (xµt+h |µh = k) =
                            k=0
                                        (ñâîéñòâî ìàðêîâîñòè)
          ∞
          X                                                        ∞
                                                                   X
      =         P (µh = k|µ0 = 1)M (xµt |µ0 = k) =                       P (µh = k|µ0 = 1)·
          k=0                                                      k=0
                          (ñâîéñòâî îäíîðîäíîñòè)

                 ·F (t, x)k = µh + (1 − (λ + µ)h)F + λhF 2 + · · · .
           dF       F (t + h, x) − F (t, x)
              = lim                         = µ − (λ + µ)F + λF 2 .
           dt   h→0            h


                                                  121

Страницы