Теория вероятностей и математическая статистика. Билялов Р.Ф. - 114 стр.

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0 =
t
0
< t
1
< ··· < t
n
∆
k
= [t
k
, t
k
+ h
k
], h
k
> 0, k = 1, ..., n,
h
k
∆
k
-` ` `` ` ` `
` `
t
1
t
1
+ h
1
t
k
t
k
+ h
k
[ ] [ ]
∆
1
∆
k
P (Θ
1
∈ ∆
1
, Θ
2
∈ ∆
2
, ..., Θ
n
∈ ∆
n
) =
= P
µ
n
\
k=1
½
(ξ
t
k
− ξ
t
k−1
= 0) ∩ (ξ
t
k
+h
k
− ξ
t
k
= 1)
¾¶
=
n
Y
k=1
e
−λ(t
k
−t
k−1
)
·
·λh
k
e
−λh
k
= h
1
···h
n
λ
n
e
−(t
1
−t
0
+t
2
−t
1
+···+t
n
−t
n−1
+h
1
+···+h
n
)
=
= λ
n
h
1
···h
n
e
−λt
n
e
−λ(h
1
+···+h
n
)
=
Z
···
Z
∆
1
×···×∆
n
p
Θ
1
....Θ
n
(t
1
, ..., t
n
)dt
1
···dt
n
,
p
Θ
1
....Θ
n
(t
1
, ..., t
n
) = λ
n
e
−λt
n
.
τ
k
= Θ
k
−Θ
k−1
F
τ
1
,τ
2
,...,τ
n
(s
1
, s
2
, ..., s
n
) = P (τ
1
< s
1
, τ
2
< s
2
, ..., τ
n
< s
n
) =
= P (Θ
1
< s
1
, Θ
2
− Θ
1
< s
2
, ..., Θ
n
− Θ
n−1
< s
n
) =
=
s
1
Z
−∞
du
1
s
2
+u
1
Z
−∞
du
2
···
s
n
+u
n−1
Z
−∞
du
n
λ
n
e
−λu
n
,
p
τ
1
,τ
2
,...,τ
n
(s
1
, s
2
, ..., s
n
) =
∂
n
∂s
1
···∂s
n
F
τ
1
,τ
2
,...,τ
n
(s
1
, s
2
, ..., s
n
) =
    Äîêàçàòåëüñòâî. Âîçüìåì ïîñëåäîâàòåëüíîñòü ìîìåíòîâ 0 =
t0 < t1 < · · · < tn è ïîëîæèì ∆k = [tk , tk + hk ], hk > 0, k = 1, ..., n,
ãäå hk òàêèå, ÷òî ∆k íå ïåðåñåêàþòñÿ.
                            ∆1                                                   ∆k
             `        [`               ]`       `      `     `              [`        ]`               -
                      t1         t1 + h1                                tk tk + hk


    Òîãäà ìîæíî ïðîâåñòè ñëåäóþùèå âû÷èñëåíèÿ

                           P (Θ1 ∈ ∆1 , Θ2 ∈ ∆2 , ..., Θn ∈ ∆n ) =
    µ\
     n ½                                       ¾¶ Yn
 =P     (ξtk − ξtk−1 = 0) ∩ (ξtk +hk − ξtk = 1)  =   e−λ(tk −tk−1 ) ·
         k=1                                                                               k=1

     ·λhk e−λhk = h1 · · · hn λn e−(t1 −t0 +t2 −t1 +···+tn −tn−1 +h1 +···+hn ) =
                                          Z     Z
= λn h1 · · · hn e−λtn e−λ(h1 +···+hn ) =   ···      pΘ1 ....Θn (t1 , ..., tn )dt1 · · · dtn ,
                                                           ∆1 ×···×∆n

îòêóäà ñëåäóåò, ÷òî pΘ1 ....Θn (t1 , ..., tn ) = λn e−λtn . Òåïåðü íàéäåì ðàñ-
ïðåäåëåíèå ñëó÷àéíûõ âåëè÷èí τk = Θk −Θk−1 . Äëÿ ýòîãî âû÷èñëèì
èõ ñîâìåñòíóþ ôóíêöèþ ðàñïðåäåëåíèÿ:

        Fτ1 ,τ2 ,...,τn (s1 , s2 , ..., sn ) = P (τ1 < s1 , τ2 < s2 , ..., τn < sn ) =

                 = P (Θ1 < s1 , Θ2 − Θ1 < s2 , ..., Θn − Θn−1 < sn ) =
                           Zs1         s2Z+u1                    sn +u
                                                                    Z n−1
                     =           du1        du2 · · ·                   dun λn e−λun ,
                           −∞          −∞                         −∞

îòêóäà, äëÿ ñîâìåñòíîé ïëîòíîñòè âåðîÿòíîñòè ïîëó÷àåì
                                                        ∂n
      pτ1 ,τ2 ,...,τn (s1 , s2 , ..., sn ) =                     Fτ ,τ ,...,τ (s1 , s2 , ..., sn ) =
                                                    ∂s1 · · · ∂sn 1 2 n



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