1. Poisson Process Exercise

Hi everyone i'm new in this forum and i wanna ask if there someone that can help me with this kind of exercises in which i have an arrival process according to a Poisson Process and life times according to an exponentially distributed random variable.

Customers arrive at a certain facility according to a Poisson process of rate /lambda. Suppose that it is known that five customers arrived in the first
hour. Each customer spends a time in the store that is a random variable, exponentially distributed with parameter /alpha and independent of the other customer times, and then departs. What is the probability that the store is empty at the end of this first hour?

2. Re: Poisson Process Exercise

This is a classical example involving the expected value of a symmetric function of arrivial times of a Poisson process, conditional on the Poisson process. See e.g. Theorem 1.2 in

http://www.maths.qmul.ac.uk/~ig/MAS3...form%20d-n.pdf

Let be the -th customer arrival time and
be the corresponding time spended in the store,

Now the required probability is

given by the theorem, in which

by iid assumption

by Law of Total Probability

by the exponential CDF

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