(Infinitely divisible distribution) A random variable X is said to be infinitely divisible if for every n, X can be written as x = x { + ••• + x n where the random variables X u...,Xn are independent...

(Infinitely divisible distribution) A random variable X is said to be

infinitely divisible if for every n, X can be written as

x = x { + ••• + x n

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.