A simple characterization of Student\u27s distributions and normal distributions

Let X1:3≤X2:3≤X3:3 be the order statistics of the independent and identically distributed random variables X1, X2, and X3. Using the regressional property E[(aX2:3-X1:3)2|X2:3=x]=E[(X3:3-aX2:3)2|X2:3=x], we characterize some distributions, including Student\u27s tν distributions (ν\u3e2) and normal distributions. © 2011