Poisson Binomial Distribution

The Poisson binomial distribution is the statistical distribution of a sum

of independent random variables having Bernoulli distributions with possibly different success probabilities . Its probability generating function is

Consequently, is the coefficient of in this product, while the expectation value and variance are and , respectively. If every is equal, the distribution reduces to the binomial distribution. When the probabilities are small, a Poisson distribution with mean can provide an approximation.