Poisson binomial distribution

In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson. In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson. In other words, it is the probability distribution of thenumber of successes in a sequence of n independent yes/no experiments with success probabilities p 1 , p 2 , … , p n {displaystyle p_{1},p_{2},dots ,p_{n}} . The ordinary binomial distribution is a special case of the Poisson binomial distribution, when all success probabilities are the same, that is p 1 = p 2 = ⋯ = p n {displaystyle p_{1}=p_{2}=cdots =p_{n}} . Since a Poisson binomial distributed variable is a sum of n independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the n Bernoulli distributions: For fixed values of the mean ( μ {displaystyle mu } ) and size (n), the variance is maximal when all success probabilities are equal and we have a binomial distribution. When the mean is fixed, the variance is bounded from above by the variance of the Poisson distribution with the same mean which is attained asymptotically as n tends to infinity. The probability of having k successful trials out of a total of n can be written as the sum where F k {displaystyle F_{k}} is the set of all subsets of k integers that can be selected from {1,2,3,...,n}. For example, if n = 3, then F 2 = { { 1 , 2 } , { 1 , 3 } , { 2 , 3 } } {displaystyle F_{2}=left{{1,2},{1,3},{2,3} ight}} . A c {displaystyle A^{c}} is the complement of A {displaystyle A} , i.e. A c = { 1 , 2 , 3 , … , n } ∖ A {displaystyle A^{c}={1,2,3,dots ,n}setminus A} . F k {displaystyle F_{k}} will contain n ! / ( ( n − k ) ! k ! ) {displaystyle n!/((n-k)!k!)} elements, the sum over which is infeasible to compute in practice unless the number of trials n is small (e.g. if n = 30, F 15 {displaystyle F_{15}} contains over 1020 elements). However, there are other, more efficient ways to calculate Pr ( K = k ) {displaystyle Pr(K=k)} . As long as none of the success probabilities are equal to one, one can calculate the probability of k successes using the recursive formula