Sum of poisson distributions

If X\sim Po(\lambda_X) and Y\sim Po(\lambda_Y), then X+Y\sim Po(\lambda_X + \lambda_Y).

What that means is that, if we have two independent discrete random variables following a poisson distribution, then their sum also follows a poisson distribution, with the mean being the sum of the means of the two original distributions.

Examples

• If X\sim Po(3) and Y\sim Po(5), then X+Y\sim Po(3 + 5) = Po(8).
• If X\sim Po(2) and Y\sim Po(4), then X+Y\sim Po(2 + 4) = Po(6).
• If X\sim Po(1) and X+Y\sim Po(2), then Y\sim Po(2 - 1) = Po(1).
• If Y\sim Po(6) and X+Y\sim Po(10), then X\sim Po(10 - 6) = Po(4).