$X \sim$ Poisson$(λ)$. What is the distribution of $X/c$? $(c > 0)$

Question

Application: $X$ is the number of particles in a closed volume. $c$ is a constant that converts from particle count to ($>0$) molar concentration. For various reasons, I want to model $Y = X/c$ instead of $X$.

I know $E(X/c) = \frac{\lambda}{c}\,$, and $\text{var}(X/c) = \frac{\lambda}{c^2}$.

The trick is that I'm moving from the integer domain to the real domain. I think the Jacobian method for transforming random variables doesn't map PMFs to PDFs, unless I'm mistaken. Haven't tried MGFs or characteristic functions.

I can use another distribution to model this, like a normal or gamma with these means and variances. But I feel like $X/c$ should have some "natural" common exponential family distribution associated with it.

Answer 1

You are not moving from the integer domain to the real domain. The support for $Y = X/c$ is discrete and given by:

$${0, 1/c, 2/c, 3/c, ...}$$

If $y$ is a real number such that $cy$ is a non-negative integer, say $cy=k$, then:

$$P[Y=y]=\frac{\lambda^k e^{-\lambda}}{k!}$$

On the other hand, for any real $y$ such that $cy$ is not a non-negative integer, $P[Y=y]=0$. So, only certain reals are in the support (only countably many).