Probability distribution problem #1

Question

Let $X_1$ and $X_2$ be i.i.d random variables having $\text{Poi}(\lambda)$ distribution. Put $T=X1+2\cdot X2$. So I have to find $\mathbb{P}(X_1+2 \cdot X_2=t)$. I can't solve this. How should I proceed.

Answer 1

The standard method to get the distribution for a sum of two independent random variables is convolution. For discrete variables this takes the form of a sum. For $X_1+2X_2=t$ an individual $(kth)$ term in the sum for $P(T=t)$ is $P(X_2=k)P(X_1=t-2k)$. The upper limit on the sum is $N=\lfloor\frac{t}{2}\rfloor$. Therefore $P(T=t)=e^{-2\lambda}\sum_{k=0}^N\frac{\lambda^k}{k!}\frac{\lambda^{t-2k}}{(t-2k)!}=\lambda^te^{-2\lambda}\sum_{k=0}^N\frac{\lambda^{-k}}{k!(t-2k)!}$. Unfortunately this last sum doesn't seem to have a neat expression.