Let $T_i$ for $i=1,2,…$ be a sequence of i.i.d exponential random variables with common parameter $\lambda$.
Let $N$ be a geometric random variable with parameter $(1/(p+1))$ that is independent of the sequence $T_i$.
Let $X$ be the sum of the $T_i$ from 1 to $N$ Show that the distribution of X is exponential.
I would like to use MGFs. I'm not sure how to incorporate the MGF of N in this case.
Best Answer
Consider a Poisson process of rate $\lambda$. Independently, each occurrence of the Poisson process is "special" with probability $q = 1/(p+1)$. Your $T$ is the waiting time until the first special occurrence.
You can also consider this from a different point of view: the special and the non-special occurrences form independent Poisson processes with rates $q\lambda$ and $(1-q)\lambda$. The waiting time until the first special occurrence is then exponential with rate $q\lambda$.