If I am given $X$ that follows an exponential distribution with mean $m$ and $Y$ that follows a poisson distribution with mean $n$, how can I use them to find the conditional probability density function of$ X$ given $Y=y$? $X$ and $Y$ are dependent.
The waiting time for a customer to be served at a bank follows an exponential distribution with mean $5$. On any particular day, the number of customers visiting the bank follows a Poisson distribution with mean $10$. Deﬁne the total waiting time $S_N = X_1 + X_2 + · · · + X_N$ where $X_i$ is the waiting time of the $i$th customer, and $N$ is the number of customers.
This is the question I'm given to find the conditional probability density function of $S_N$ given $N=n$.