Suppose I have a variable, $Y$, that is a scaled Poisson random variable. That is:
$$ Y = kX $$
and
$$ X \sim \mathrm{Poisson}(\mu) $$
This means that PMF of $Y$, $P(y)$, is given by:
$$ P(y) = e^{-\mu} \frac{\mu^{\frac{y}{k}}}{\left(\frac{y}{k}\right)!} $$
What would the mean and variance of $Y$ be in this situation? (Of course, the mean and variance of $X$ is just $\mu$.)
Best Answer
Close. Always include the support in any pmf.$$\mathsf P(y) = \dfrac{\mu^{\frac{y}{k}}e^{-\mu}}{\left(\tfrac{y}{k}\right)!}\mathbf 1_{y\in k\Bbb Z^+} $$
$k\Bbb Z^+=\{kx: x\in\Bbb Z. x\geq 0\}$
Scaled.
$\begin{split}\mathsf E(Y) &= \mathsf E(kX)\\[1ex] & =k\mu\\[2ex]\mathsf {Var}(Y)&=\mathsf{Var}(kX)\\[1ex]&=k^2\mu\end{split}$