Convolution of Two Independent Poisson Random Variables

Let $X \sim \text{Poisson}(2)$ and $Y \sim \text{Poisson}(3)$, independent.

(a) Compute $P(X + Y = 3)$ using the convolution formula $P(X+Y=k) = \sum_{j=0}^{k} P(X=j)P(Y=k-j)$.

(b) Verify your answer using the fact that $X + Y \sim \text{Poisson}(5)$.

(c) Compute $P(X = 1 \mid X + Y = 3)$.