# Probability of discrete-random process

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 $$P[W>w]$$.

• $$P[W>w] = P[N_w=0] = P_{N_w}(0)$$
• So you can find $$P[W\le w]$$ as a cumulative distribution function
In fact this is a Poisson process with a rate of $$5$$ per minute. $$W$$ has an exponential distribution with the same rate