Find conditional distribution for Poisson random variables

Question

Suppose that random variables $X \in P(\lambda)$ and $Y \in P(\mu)$ are independent. I need to find the conditional distribution of random variable X given that $X+Y=n,$$n \ge1.$ I'm not sure how to start on this problem, also how would I use the independence of the two variables when solving?

Answer 1

For $0 \le x \le n$, \begin{align} P(X=x \mid X+Y=n) &= \frac{P(X=x,X+Y=n)}{P(X+Y=n)} \\ &= \frac{P(X=x,Y=n-x)}{P(X+Y=n)} \\ &= \frac{P(X=x) \cdot P(Y=n-x)}{P(X+Y=n)}, \end{align} where the last equality is due to independence of $X$ and $Y$. To compute the denominator, you need to use the fact that $X+Y \sim P(\lambda+\mu)$. This requires justification, but it is probably in your textbook/notes.