PROBABILITY GENERATING FUNCTIONS OF ABSOLUTE DIFFERENCE OF TWO RANDOM VARIABLES

Communicated by Jerzy Neyman, August 1, 1966 1. All random variables considered in this note are nonnegative and integervalued. Generically, the probability generating functions (P. G. F.) of such variables are denoted by G with subscripts identifying the random variables concerned. The argument, or arguments of the P. G. F., denoted by either u or v, will be assumed to have their moduli less than unity. The note gives a simple formula for the P. G. F. of the absolute difference Z = i X2|, where X1 and X2 are arbitrary random variables of the kind considered. Later, this formula is used to obtain a characterization of the geometric distribution. The random variable Z appears in several domains of applications. See, for instance, David.1 2. THEOREM 1. Whatever be the random variables X1 and X2, we have = 1 (2I 1 -V2 *GS x(eiGe i)dO. (1) i(V) 2~r Jo 1+ v2-2v cosO 1, The theorem can be easily proved by first noticing that for vI < 1, 1 V2 co