Does the following constitute a proof that the multiplication of a Poisson random variable $K$ with an integer constant $a$ is not itself Poisson? That is,
$f_K(k) = \frac{\lambda^k}{k!} e^{-\lambda}$
$L = aK$
Does not imply $L$ is a Poisson random variable.
Intuitively, multiplication of an integer-valued distribution leaves "gaps" on the number line, which would mean $aK$ cannot be poisson.
Here is my attempted proof:
$E[K] = \lambda$
$E[L] = a\lambda$
$Var(L) = a^2 \lambda$
If $L$ is Poisson, then $E[L] = Var(L)$
Which is a contradiction. Additionally, does that mean that $L$ is poisson iff. $a = 1$?