Let $X_{1}, \dots, X_{n}$ be independent Poisson random variables (parameter $\lambda$), and set $Y=\sum_{i=1}^{n}X_{i}$. Find the conditional probability mass function of the random variable $X_{i}$ given $Y=m$ for an integer $m$.
I have done the first part of the problem, where the distribution of the random variable $Y$ will be given by a Poisson random variable of parameter $n\lambda$ by using the mgf technique to identify the distribution of $Y$. However, I don't know how to compute $P(X_{i}=x | Y=m)$ since I don't know how to compute the joint pmf of $(X_{i},Y)$ (else, this should be straightforward).
How would one proceed with the problem? I initially though that $X$ and $Y$ could be independent, but I don't think that's the case.
Best Answer
Outline: We want to compute $\Pr(X_i=k\mid Y=m)$. By the definition of conditional probability, this is $$\frac{\Pr((X_i=k)\cap (Y=m))}{\Pr(Y=m)}.\tag{1}$$ Note that $0\le k\le m$. You know how to compute the denominator, so we concentrate on the numerator.
The numerator is $$\Pr(X_i=k)\Pr(W=m-k),$$ where $W$ is the sum of all the $X_j$ except for $X_i$.
You know how to compute $\Pr(W=m-k)$.
Now when you substitute in (1) there will be a pleasant amount of cancellation.