I'm stuck proving the following proposition:
Let $\{E_i\}_{i\geq 1}$ be i.i.d. exponential random variables on $[0,\infty)$ with parameter $1$: $P(E_i > x)= e^{-x}, x>0.$ Let $\Gamma_{n} =\sum_{i=1}^{n} E_i,n\geq 1.$ Show that $$N=\sum_{n\geq 1}\delta_{\Gamma_{n}},$$ is a homogeneous Poisson process on $[0,\infty).$
Because of the hypotesis, $\Gamma_{n}$ has distribution $\Gamma(n,1).$ Next, I tried to compute the "rate" or mean of the process, but I found with a hard series to calculate.
Any kind of help is thanked in advance.