[機統] poisson process

[nourri (nourri)]

Starting at some fixed time, satellites are launched at times of a Poisson
process with rate lambda. After an independent amount of time having
distribution functino F and mean mu, the satellite stops working. Let X(t) be
the number of satellites working at time t.

Find

a)the distribution of X(t).

b) as t goes to infinity, show the limiting distribution is poisson(lambda*mu)

http://tinyurl.com/oznez4r

http://tinyurl.com/qb6kghw

上面是talkstats跟mathhelpforum的答案

可是要怎麼證明P(Xi+Ti>t)=(1/t)*E(min(Ti,t))

where Ti is the lifespan of i-th satellite (which launched at time Xi)

第一個連結有P(Xi+Ti>t)的答案

但是我看不出來把1/t提出來後剩下那一部分是E(min(Ti,t))

所以想請教一下板友

謝謝

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→ nourri      :我做change of variable最後得到E(Ti)as t->infinity