I have the following question for homework and I'm not sure how to get it started. It says to suppose that $N$ is a Poisson random variable with parameter $\mu$. Given $N=n$, random variables $X_1,X_2,X_3,\ldots,X_n$ are independent with uniform $\sim (0,1)$ distribution. So there are a random number of $X$'s. Given $N=n$ what is the probability that all the $X$'s are less than $t$?
So I set up the problem as I'm looking for: $\begin{align}P(X\lt t\mid N=n) =\frac{P(X\lt t, N=n)}{P(N=n)}\end{align}$
I'm unsure of how to find the joint density function for the numerator. Since they aren't independent, because the probability of $X$ depends on $N$. I'm also unsure of the $t$ as well, so would this mean that if $X$ was unconditioned that$P(X_j\lt t)=\frac{1}{1-t}$??