Let $X$ is poisson distribution.
$$f_{X}(n;\lambda)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$
And there is some positive constant $\alpha$.
I like to know pmf(probability mass function) of $Z=\alpha X$.
I have searched this topic and i could find almost similar post.
This post deal with distribution of $Y=\frac{X}{m}$ ($X$ is poisson and $m$ is constant).
But it is not easy to understand to me.
Especially calculating pgf(probability generating function) of $Y$ part!
What is $D_{m,t}p(x)$?
Please explain more detail.
Best Answer
If $P(X=n) = \dfrac{\lambda^{n}}{n!}e^{-\lambda}$ when $n$ is a non-negative integer and $0$ otherwise, and $Z=\alpha X$ then $$P(Z=z)=\dfrac{\lambda^{z / \alpha}}{(z/\alpha)!}e^{-\lambda}$$ when $z/\alpha$ is a non-negative integer and $0$ otherwise