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If $N_{\tau}$ is random process of number of items sold in $\tau$ minutes by probability below

$$ \begin{align*} P_{N\tau}(n)=(5\tau)^n e^{-5τ}/n! \end{align*} $$ for $n=0,1,2,...$

Then imagine you wait a random time of $W$ minutes until you see an item is sold. What is the distribution of W?

I know that it asks for $P[W>w]$.

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Hints:

  • $P[W>w] = P[N_w=0] = P_{N_w}(0)$

  • So you can find $P[W\le w]$ as a cumulative distribution function

  • and its derivative as a probability density function

In fact this is a Poisson process with a rate of $5$ per minute. $W$ has an exponential distribution with the same rate

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