A simple characterization of Student's t_2 distribution

摘要

We consider independent random variables X, X_1, X_2, X_3 with a common continuous distribution function F(x) and a finite mean E(X) = a, Let X_(1:3)≤ X_(2:3)≤ X_(3:3) be the order statistics produced by the random varibles X_, X_2 and X_3. In this paper, we characterize the distributions for which the conditional expectation E (X_1X_2X_3|x_(2:3)) is a linear function of x. Some simple properties of the regression function are shown to characterize Student's t_2 distribution.