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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$$P(X_j\lt t)=\frac{1}{1-t}$$??

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1 Answer 1

up vote 1 down vote accepted

Given that $N=n$, we don't need to worry about the fact that $N$ is Poisson, we actually know $n$. (Presumably this is the beginning of a more elaborate problem in which the distribution of $N$ will be relevant.)

For fixed $n$, calculating the probability is, I imagine, easy for you.

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