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.
Citation
E. M. Bolger. W. L. Harkness. "Characterizations of Some Distributions by Conditional Moments." Ann. Math. Statist. 36 (2) 703 - 705, April, 1965. https://doi.org/10.1214/aoms/1177700179
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