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 $a_1, a_2 \in \mathcal{R}$?
Best Answer
If $X \sim Poisson(\lambda)$ and $Y=cX$, then $\mathbf{E}Y = c \lambda \neq \mathbf{Var} Y = c^2 \lambda$.