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Let $\{N(t) : t \geq 0\}$ be a Poisson process with rate $λ$, and $Z$ represent the number of arrivals in the interval of time $[0,t]$. Let $T$ be a random variable, exponentially distributed with parameter $µ > 0$, independent of $N(t)$. Determine the distribution of the number of arrivals $Z$, happening in the random interval of time $[0,T]$.

Any hints?

Ilham
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user220502
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1 Answers1

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The distribution is also Poisson. Since this process is a Compound Poisson process. It's mean is $\lambda$TE[T] and it's variance is $\lambda$TE[$T^2$].

Here's a good reference:

http://www.columbia.edu/~ww2040/3106F14/lec1023.pdf

Ragnar
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