Chapter 1 - Introduction

Abstract

Laha and Kotlarski gave a complete description of the family of all density functions f such that the quotient X/Y follows the standard Cauchy distribution whenever X and Y are independent and identically distributed (i.i.d.) with density f. A natural question is to find additional conditions under which the normal distribution can be identified from the distribution of quotients of independent random variables. Kotlarski (1967) proved that if X, Y, and Z are independent real-valued random variables with density functions symmetric about zero, where U = X/Z and V = Y/Z, then X, Y, and Z are normally distributed with a common variance σ 2. Even though the distribution of the ratio U = X/Y of two independent random variables X and Y does not determine the distributions of X and Y, the situation changes completely if the joint distribution of two ratios is considered as U = X/Z and V = Y/Z, where X, Y, and Z are three independent random variables. Kotlarski's result indicates that if the joint distribution is bivariate Cauchy, then X, Y, and Z are normally distributed under some technical assumptions.