Characterizations of Some Distributions by Conditional Moments

Abstract

Let $X_1$ and $X_2$ be independent random variables (r.v.'s) and assume that $Y = X_1 + X_2$ has finite second moment. We assume that the mean and variance of $X_1$, conditional on fixed values $y$ of $Y$, satisfy the structural relations $(i) E(X_1 \mid Y = y) = \lambda_1y/\lambda\quad\text{and} (ii) V(X_1 \mid Y = y) = (\lambda_1\lambda_2/\lambda^2)u(y)$ where $\lambda_1$ and $\lambda_2$ are positive constants, $\lambda = \lambda_1 + \lambda_2$, and $u(y)$ is non-negative. Laha [2] has given a simple necessary and sufficient condition for the regression $E(X_1 \mid Y = y)$ to be linear, as we assume in (i). We use the added condition (ii) to determine explicitly the distribution functions (d.f.'s) of $X_1$ and $X_2$ (and hence of $Y$) for various choices of $u(y)$. We prove in Section 2 a theorem on which our characterizations are based and illustrate the theorem in Section 3.