PoissonBinomialverteilung

The Poisson-Binomial distribution is a probability distribution that generalizes the binomial distribution. While the binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success, the Poisson-Binomial distribution allows for different probabilities of success across trials. This makes it particularly useful in scenarios where the trials are not identically distributed.

The Poisson-Binomial distribution is defined as follows: let X1, X2, ..., Xn be independent Bernoulli random variables with probabilities of success p1, p2, ..., pn, respectively. The Poisson-Binomial distribution describes the distribution of the sum S = X1 + X2 + ... + Xn. The probability mass function of S is given by:

P(S = k) = Σ (over all subsets of size k) ∏ (over all elements in the subset) pi ∏ (over all elements not in the subset) (1 - pi)

where the summation is over all subsets of size k from the set {1, 2, ..., n}.

The Poisson-Binomial distribution is particularly useful in applications such as network reliability, where the success probabilities of different components may vary, and in quality control, where the probability of a defect may differ across items. It is also relevant in fields like genetics and epidemiology, where the probabilities of different events may not be uniform.

Despite its flexibility, the Poisson-Binomial distribution can be computationally intensive to work with, especially for large n. In practice, approximations and computational methods are often used to handle such cases. One common approximation is the Poisson approximation, which can be applied when the number of trials is large and the probabilities of success are small.