Теория вероятностей и математическая статистика. Билялов Р.Ф. - 23 стр.

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P
1
(t), P
2
(t) t
P
k
(t)
π
k
= lim
t→∞
P
k
(t); k = 0, 1, 2.
P
0
(t + h) = P
0
(t)P ( )+
+P
1
(t)P ( )+
+P
2
(t)P ( ) =
P
0
(t)(1 − (α
1
+ α
2
)h + ···) + P
1
(t)(βh + ···) + P
2
(t)(βh + ···) =
= P
0
(t) + h[−(α
1
+ α
2
)P
0
(t) + βP
1
(t) + βP
2
(t) + ···].
P
0
0
(t) = lim
h→0
P
0
(t + h) − P
0
(t)
h
= −(α
1
+ α
2
)P
0
(t) + βP
1
(t) + βP
2
(t).
P
0
1
(t) = α
1
P
0
(t) − βP
1
(t) + α
1
P
2
(t),
P
0
2
(t) = α
2
P
0
(t) − (β + α
1
)P
2
(t).
P
i
(t) → π
i
, π
0
+ π
1
+ π
2
= 1.
−(α
1
+ α
2
)π
0
+ β(π
1
+ π
2
) = 0, α
2
π
0
− (β + α
1
)π
2
= 0 ,
π
0
=
β
α
1
+α
2
+β
, π
1
=
α
1
α
1
+β
, π
2
=
α
2
α
1
+β
π
0
.
t
1
t
2
(t
2
> t
1
),
t
1
t
2
.
3
4
.
A
B =
P (B|A) P (A|B).
P1 (t), P2 (t)  âåðîÿòíîñòè òîãî, ÷òî â ìîìåíò t ëèíèÿ ñâîáîäíà, çàíÿ-
òà ñðî÷íûì âûçîâîì, çàíÿòà ïðîñòûì âûçîâîì. Íàïèñàòü äëÿ Pk (t)
äèôôåðåíöèàëüíûå óðàâíåíèÿ è íàéòè πk = lim Pk (t); k = 0, 1, 2.
                                                               t→∞
   Ðåøåíèå. Ïðèìåíÿÿ ôîðìóëó ïîëíîé âåðîÿòíîñòè, ìîæåì çàïè-
ñàòü:
                  P0 (t + h) = P0 (t)P (âûçîâ íå ïîñòóïèë)+
                  +P1 (t)P (ñðî÷íûé âûçîâ ïðåêðàòèëñÿ)+
                  +P2 (t)P (ïðîñòîé âûçîâ ïðåêðàòèëñÿ) =
  P0 (t)(1 − (α1 + α2 )h + · · · ) + P1 (t)(βh + · · · ) + P2 (t)(βh + · · · ) =
         = P0 (t) + h[−(α1 + α2 )P0 (t) + βP1 (t) + βP2 (t) + · · · ].
              P0 (t + h) − P0 (t)
 P00 (t) = lim                    = −(α1 + α2 )P0 (t) + βP1 (t) + βP2 (t).
          h→0          h
                      P10 (t) = α1 P0 (t) − βP1 (t) + α1 P2 (t),
                        P20 (t) = α2 P0 (t) − (β + α1 )P2 (t).
 ïðåäåëå Pi (t) → πi , π0 + π1 + π2 = 1. Ïîëó÷àþòñÿ óðàâíåíèÿ

        −(α1 + α2 )π0 + β(π1 + π2 ) = 0, α2 π0 − (β + α1 )π2 = 0,
                     β                 α1               α2
îòêóäà π0 =      α1 +α2 +β ,   π1 =   α1 +β ,   π2 =   α1 +β π0 .

                        Çàäà÷è äîìàøíåãî çàäàíèÿ.
   Çàäà÷à 14.165. Âåðîÿòíîñòü òîãî, ÷òî ïðèáîð íå îòêàæåò ê ìî-
ìåíòó âðåìåíè t1 , ðàâíà 0.8, à âåðîÿòíîñòü òîãî, ÷òî îí íå îòêàæåò
ê ìîìåíòó âðåìåíè t2 (t2 > t1 ), ðàâíà 0.6. Íàéòè âåðîÿòíîñòü òîãî,
÷òî ïðèáîð, íå îòêàçàâøèé ê ìîìåíòó âðåìåíè t1 , íå îòêàæåò è ê
ìîìåíòó âðåìåíè t2 .

                                                                       Îòâåò: 34 .

   Çàäà÷à 14.169. Ïîäáðàñûâàþò íàóäà÷ó òðè èãðàëüíûå êîñòè.
Íàáëþäàåìûå ñîáûòèÿ: A={íà òðåõ êîñòÿõ âûïàäóò ðàçíûå ãðàíè},
B ={õîòÿ áû íà îäíîé èç êîñòåé âûïàäåò øåñòåðêà}. Âû÷èñëèòü
P (B|A) è P (A|B).

                                                23

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