PoissonBinomial

The Poisson-Binomial distribution is a discrete probability distribution that describes the sum of independent but not necessarily identically distributed Bernoulli random variables. In simpler terms, it's the distribution of the number of successes in a series of independent trials, where each trial can have a different probability of success. This is a generalization of the Binomial distribution, where all trials have the same probability of success.

The probability mass function (PMF) of a Poisson-Binomial distribution is given by a convolution of the individual Bernoulli probabilities. If X_i is a Bernoulli random variable with probability of success p_i, and X = sum(X_i) for i=1 to n, then X follows a Poisson-Binomial distribution. Calculating the PMF directly can be computationally intensive, especially for a large number of trials.

Applications of the Poisson-Binomial distribution arise in scenarios where events occur with varying probabilities. Examples include modeling the number of defective items in a batch where defect rates differ across production lines, or in risk analysis where different assets have independent probabilities of failure. While direct calculation of the PMF can be complex, approximations or recursive methods are often employed for practical analysis. The mean of a Poisson-Binomial distribution is the sum of the individual probabilities of success, and its variance is the sum of the individual variances.