I know that for a Pois$(\lambda)$ distribution, the expectation is $\lambda$. I also know its relation to the exponential distribution, and how you can derive the PMF of the exponential through the Poisson. With this, it's pretty easy to calculate the PMF and derive the expectation as $\frac{1}{\lambda}$
My question is, how you can formally prove the expectation of an exponential distribution, just using the fact it's a waiting time between Poisson points? I believe it must be possible because the expectation makes intuitive sense; if a poisson process has 3 calls/hr, then the average waiting time intuitively should be 1/3 hrs. I'm just not able to formalize this. Thanks!