Compound Probability Distribution with Examples

A compound probability distribution is the distribution of some combination of independent and identically distributed variables whose parameters themselves follow some probability distribution.

Compound Poisson Distribution:

Suppose X₁,…, X_N are independent and identically distributed Poisson variables with N itself being a random variable following Poisson distribution then,

Y = X₁+….+X_N is said to follow compound Poisson distribution.

Then mean of this distribution is given as E(Y) =E(N)*E(X_i)

The compound Poisson distribution finds applications in the theory of stochastic processes and queuing theory.

Other examples of compound distributions are:

1. A normal distribution whose mean follows normal distribution is once again normally distributed.
2. A Poisson distribution whose parameter is distributed according to a gamma distribution gives a negative binomial distribution.

Mixture Distribution:

If we are given a collection (possibly uncountable) of random variables and we choose one of them randomly according to some probabilistic method then the resulting distribution so obtained is called a mixture distribution.