Theorem: Let {N(t): t≥0} be a Poisson process of rate λ. Suppose we are given that for a fixed t, N(t)=n. Let T(adsbygoogle = window.adsbygoogle || []).push({}); _{i}be the time of the i^{th}event, i=1,2,...n.

Then the (conditional) density function of T_{n}given that N(t)=n is the exactly the same as the density function of X_{(1)}=min{X_{1},X_{2},...,X_{n}}, where X_{1},X_{2},...,X_{n}are i.i.d. uniform(0,t).

(similar result holds for the density of the other T_{i}'s)

Example: Let {N(t): t≥0} be a Poisson process of rate λ. The points are to be thought of as being the arrival times of customers to a store which opens at time t=T. The customers arriving between t=0 and t=T have to wait until the store opens. Let Y be the total times that these customers have to wait. Calculate E(Y).

Solution:

N(T)=N

N(T)~Poisson(λT)

(T_{1},T_{2},...,T_{N}) is equal in distribution to (X_{(1)},X_{(2)},...,X_{(N)}), where the order statistics are coming from X_{1},X_{2},...,X_{N}which are i.i.d. uniform(0,T).

=> T_{1}+T_{2}+...+T_{N}is equal in distribution to X_{(1)}+X_{(2)}+...+X_{(N)}= X_{1}+X_{2}+...+X_{N}

E(Y)=E(total waiting time)

=E[(T-T_{1})+(T-T_{2})+...+(T-T_{N})]

=E(NT) - E(T_{1}+T_{2}+...+T_{N})

=T E(N) - E(X_{1}+X_{2}+...+X_{N})

=T E(N) - E[E(X_{1}+X_{2}+...+X_{N}|N)]

=T E(N) - E[N E(X_{1})]

=T E(N) - E[N T/2]

=T E(N) - (T/2) E[N]

=(1/2) T E(N)

=(1/2) T λT

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In the theorem, we require N(t)=n where n is a FIXED number. But throughout the solution (for example when they calculate E[X_{1}+X_{2}+...+X_{N}]), N(t)=N is being treated as a RANDOM VARIABLE rather than a fixed number. Why? If N is a random variable, I don't think the theorem above applies...will we still have that T_{N}is equal in distirbution to X_{(N)}? Why or why not?

In other words, I am questioning the statement "(T_{1},T_{2},...,T_{N}) is equal in distribution to (X_{(1)},X_{(2)},...,X_{(N)})" in the solution of the example. Here N itself is a random variable, but in the theorem it is required to be fixed (in the theorem we're GIVEN the value of N(t))...

Can someone please explain? Any help is greatly appreciated!:)

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# Poisson counting process & order statistics

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