(Infinitely divisible distribution) A random variable X is said to be infinitely divisible if for every n, X can be written as

where the random variables X u...,Xn are independent and have the same distribution function Fn( X ) (depending on n). The distribution function of an infinitely divisible random variable is called the infinitely divisible distribution function. Show that normal, gamma, Cauchy, and Poisson distributions are infinitely divisible distributions.

+ X