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1 $begingroup$ Problem: I have a sum of $N$ random variables $X_i$ , where $N$ is distributed according to a binomial distribution and the $X_i$ are independent and identically distributed according to an exponential distribution. I would like to get the pdf of the sum $$ Y_N = X_1 + X_2 + dots + X_N. $$ If I write the pmf of $N$ as $$ p_n = {N choose n} p^n q^{N-n}$$ and the pdf of the $X_i$ as $$ f(x) = lambda e^{-lambda x},$$ How can I get the pdf $p(y)$ of $Y_N$ ? Attempt: I gather this is called a compound process. I read about it somewhat in Feller: the section is "Sum of a Random Number of Random Variables", but he does not mix discrete and continuous random variables. I think this is just a discrete time random walk where the steps are exponentially distributed. I think I ...