May I ask a question regarding compound random variables, in which I want to obtain the distribution of $T$? $T=\sum_{i=1}^{N}t_i$, where $t_i$ are i.i.d exponential random variables with parameter $\lambda$, and $N$ is geometric distributed random variable, i.e. $P(N = k) = (1 - p)^kp$ . What I have tried now, is as follow
since $\sum_{i = 1}^n t_i $ follows gamma distribution with parameter $n, \lambda$
$P(T = t) = P(\sum_{i=1}^N t_i = t) = \sum_{i = 1}^\infty P(\sum_{i = 1}^n t_i = t) P(N = n)$
after that, when I try to simplify the formula using gamma distribution, I have nowhere to go, anyone can help me?
