Recent Developments in the Characterizations of the Normal Distribution

 

M. Ahsanullah (1991). Recent Developments in the Characterizations of the Normal Distribution. Journal of Statistical Research, Vol. 25, No. 1-2, pp.  19-25.

 

Abstract

Suppose X_1, X_2,\ldots,X_n are n ( \geq 2) non degenerate random variables with L_t = a_1X_1 +a_2X_2 + \ldots + a_nX_n) 0<|a_i|\lt \infty, i= 1,\ldots,n and W = (L_1 -E(L_1))^2/Var(L_1) for Var(L_1)\lt\infty. If X's are normally distributed then W is distributed as chi-squared with one degree of freedom. However if W is distributed as chi-squared with one degree of freedom then X's need not be normally distributed. In this paper some characterizations of the normality of the X's are given when W is distributed as chi-squared with one degree of freedom.

 

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