On Characterizing the Chi Square Distribution by the Student Law
Ignacy Icchak Kotlarski · Journal of the American Statistical Association · 1966
It is known that if X 0, X 1 are independent random variables identically distributed according to the chi square distribution with n degrees of freedom then the random variable follows the Student distribution with n degrees of freedom. But the chi square distribution can not be characterized by this property, because there exist pairs X 0, X 1 of independent positive random variables identically distributed according to distributions differing from chi square, and for which Y follows the Student law. Let X 0, X 1, X 2 be three independent positive random variables and let (Y 1, Y 2) be given by the formulae The aim of this paper is to show that although the distribution of one random variable Y does not characterize the chi square distribution of X 0, X 1, the distribution of the pair (Y 1, Y 2) does characterize the chi square distributions of all the three random variables X 0, X 1, X 2.