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Given a scaled Poisson random variable $Y = cX$, where $X$ is a Poisson random variable with mean $\lambda$, what is the probability mass function (PMF) of $Y$?
The PMF of $X$ is $$p(x) = e^{-\lambda} \lambda^x/x!$$ and MATLAB has a function for it, which is poisspdf. The question is how to compute the PMF of $Y$?
To avoid confusion, use subscripts to denote the corresponding random variable.
Let $y\in\{0,c,2c,\ldots\}$ and observe that when $c\ne 0$,
$$p_Y(y) = p_X\left(\{x\mid cx=y\}\right) = p_X\left(\{y/c\}\right) = e^{-\lambda}\frac{\lambda^{y/c}}{(y/c)!}.$$
When $c=0$ the only possible value of $y$ is $0$ and $$p_Y(0) = p_X\left(\{x\mid 0x=0\}\right) = p_X\left(\{0,1,2,\ldots\}\right) = 1.$$
The general rule applied here is that when $X$ is any random variable, $f:\mathbb{R}\to\mathbb{R}$ is a measurable function, and $Y=f(X)$, then for any Borel set $\mathcal{B}\subset\mathbb{R}$
$$\Pr(Y\in\mathcal{B}) = \Pr(f(X)\in\mathcal{B}) = \Pr(X\in f^{-1}(B))$$
where
$$f^{-1}(\mathcal B) = \left\{x\in\mathbb{R}\mid f(x)\in\mathcal{B}\right\}.$$
These aren't really facts about probabilities per se: you can see they are merely stating basic properties of functions.
In this particular case the map is $f(x)=cx$ and $\Pr$ is a Poisson distribution; but exactly the same approach works for any (measurable) $f$ and any distribution whatsoever.
