Linear combination of independent poisson random variables

Question

We know that if $X_1$ and $X_2$ are independent random variables such that $ X_1 \sim \text{Poisson}(\lambda_1) $ and $X_2 \sim \text{Poisson}(\lambda_2)$ that $X_1+X_2 \sim \text{Poisson}(\lambda_1+\lambda_2)$

Is there any result about a linear combination of two independent poisson random variables $a_{1} X_1+a_2 X_2$ where $a_1, a_2 \in \mathcal{R}$?

Answer 1

If $X \sim Poisson(\lambda)$ and $Y=cX$, then $\mathbf{E}Y = c \lambda \neq \mathbf{Var} Y = c^2 \lambda$.

Answer 2

$$\mathbb{E}e^{t(a_1X_1+a_2X_2)}=\frac{1}{1-\frac{t}{\lambda/a_1}}\frac{1}{1-\frac{t}{\lambda/a_2}}=\frac{1}{1-a_1/a_2}\frac{1}{1-\frac{t}{\lambda/a_1}}+\frac{1}{1-a_2/a_1}\frac{1}{1-\frac{t}{\lambda/a_2}}$$Taking the inverse Laplace transform you'll find the pdf of $a_1X_1+a_2X_2$.