Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.