Let $A_m$ be the sum of $m$ identically distributed random variables that are independent and that have an exponential distribution with parameter $\mu$. How do I prove that $A_m$ has a gamma distribution with parameters $m$ and $\mu$? And how do I prove that $M_s=\max\{m:A_m\leq s\}$ has a Poisson distribution?
I don't really know how to tackle such a problem.
The characteristic function of an exponential($\lambda$) distribution is $\frac{ \lambda}{\lambda-i \omega}$, so the sum of $n$ i.i.d. Exp($\lambda$) distributions has characteristic function $(\frac{ \lambda}{\lambda-i \omega})^n$ since sums of independent rv's have their characteristic functions multiply. Then, you can recognize this as a gamma distribution with appropriate parameters by comparing to the form of a Gamma distribution (write $\frac{ \lambda}{\lambda-i \omega} = 1+\frac{ i \omega}{\lambda-i \omega}$ will make the comparison easier).
As for the latter, think of a Poisson process.
