# How to connect Poisson distribution with exponential distribution

I have following problem to solve :

There are $$N$$ customers waiting in the checkout line, where $$N$$ has Poisson distribution with parameter $$3$$. If a service begins when $$m$$ customers are in the line, that service time has exponential distribution with parameter $$\frac{1}{2m}$$ . Find the probability that a customer just approaching the checkout will be served in less than a minute.

What I did :

If X is a time serving, then

$$P(X<1)=F(1)=1-e^{-\frac{1}{2N}}$$

according to CDF for exponential distribution. But now I don't know what to do, how to use fact that $$N$$ has Poisson distribution. I'm puzzled. How to connect these two things?

Use conditional probability. Let $$N\sim \text{Poisson}(3)\\ X|N\sim \text{Exponential}(1/(2N))$$
Then from total probability $$P(X<1)=\sum_{N=0}^\infty F(1|N)P(N)$$
$$\begin{split}\sum_{N=0}^\infty (1-e^{-\frac 1{2N}})\frac{e^{-3}3^N}{N!}&=\sum_{N=0}^\infty \frac{e^{-3}3^N}{N!}-e^{-3}\sum_{N=0}^\infty \frac{e^{-\frac 1{2N}}3^N}{N!}\\ &=1-???\end{split}$$
• thanks for your answer. I did it and got $P(X<1)=F(1|1) \cdot P(1)+F(1|2) \cdot P(2)+...= (1-e^{-\frac{1}{2}}) \cdot 3e^{-3} +(1-e^{-\frac{1}{4}}) \cdot \frac{9e^{-3}}{2} + ...$ and I don't know what's next, how to calculate it till the end. Does it even converge? Jun 10, 2021 at 16:51
• @Stacker : the series goes from 1 and not from 0. Calculating it manually ($\sum_1^9$ is enough) I got $p\approx 17.82\%$ Jun 12, 2021 at 6:18