Basic Poisson Question

Question

Just a quick question the $\mathbb{P}(X_1+2X_2=t)$ for $X_i = P_o(\lambda)$ would that just be a poisson ditribution with parameter $3\lambda$??

Answer 1

If $W=X_1 + 2 X_2$, then $\mathbf{Var} W = \mathbf{Var}X_1 + 4 \mathbf{Var} X_2$ (assuming both rvs are independent, so the covariance is 0). So $\mathbf{Var} W = 5 \lambda \neq \mathbf{E}W = 3 \lambda$, so $W$ can't be Poisson.

Answer 2

We assume the $X_i$ are independent. Note that $$\Pr(X_1+2X_2=0)=\Pr(X_1=0)\Pr(X_2=0)=e^{-2\lambda}.$$ Thus if $X_1+2X_2$ is Poisson with parameter $\mu$, then $\mu=2\lambda$.

But $$\Pr(X_1+2X_2=1)=\Pr(X_1=1)\Pr(X_2=0)=\lambda e^{-2\lambda}.$$ If $X_1+2X_2$ is Poisson with parameter $\mu$, then $\mu e^{-mu}=\lambda e^{-2\lambda}$. Together with $\mu=2\lambda$ this gives $\mu e^{-\mu}=\frac{\mu}{2}e^{-\mu}$. This is possible only in the degenerate case $\lambda=0$.