# Relationship between these two probability mass functions.

If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows:

$P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i}$

what can I infer from this equation? Any observations or interesting properties that you see based on this relation?

What if,
P($X=x_i$) = $\lambda\sqrt{\frac{P (Y=x_i)}{x_i} }$

In both cases, $\lambda$ is a constant.

## 2 Answers

(EDIT: This answer partially overlaps some comments on math.SE mentioned by Nate, see here.)

The second transformation has no interpretation that I am aware of. Could you provide some context?

The first transformation (from $X$ to $Y$) is called size-biasing. It is equivalent to ask that $E(\varphi(Y))=E(X\varphi(X))/E(X)$ for every test function $\varphi$ (for every bounded measurable $\varphi$, if you like). As such it can be defined for every nonzero integrable distribution of $X$ on the positive real halfline, discrete or continuous or otherwise. (Note that if one does not assume that $X\ge 0$ almost surely and that $E(X)$ is positive and finite, your condition makes no sense.) In particular, as noted by Suresh, your $\lambda$ is $E(X)$, but also $E(Y^k)=E(X^{k+1})/E(X)$ for every nonnegative $k$, the Laplace transform of the distribution of $Y$ is related to the first derivative of the Laplace transform of the distribution of $X$, and so on.

There are some beautiful uses of this transformation in the context of branching processes, as explained in a forthcoming book Probability on Trees and Networks by Russell Lyons with Yuval Peres.

I'm not sure how interesting this is, but in the first case, straight multiplication yields that $E[X] = \lambda$, and $E[Y] = E[X^2]/\lambda$, which allows you to bound the second moment of $X$ in terms of $E[Y]$.