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

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.

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).
• Yep. The gap between atoms of probability has simply changed from $1$ to $1/c$; it's still discrete. – Glen_b Nov 5 '16 at 1:34
• @osazuwa -- The usual methods of transformation for discrete random variates work as they should. Indeed $X/c$ does have an exponential family distribution associated with it -- it's scaled Poisson. You should be able to model that in GLM programs using the quasi-Poisson distribution. If $c$ is known you can supply the correct dispersion (at least you can in R but I assume most packages will do something similar) before producing the table of coefficients, and it should adjust the standard errors appropriately. Or the necessary calculations can be done directly from the count-GLM. – Glen_b Nov 5 '16 at 1:40