Support k ∈ { 0, …, n } | ||
Parameters p ∈ [ 0 , 1 ] n {\displaystyle \mathbf {p} \in [0,1]^{n}} — success probabilities for each of the n trials pmf ∑ A ∈ F k ∏ i ∈ A p i ∏ j ∈ A c ( 1 − p j ) {\displaystyle \sum \limits _{A\in F_{k}}\prod \limits _{i\in A}p_{i}\prod \limits _{j\in A^{c}}(1-p_{j})} CDF ∑ l = 0 k ∑ A ∈ F l ∏ i ∈ A p i ∏ j ∈ A c ( 1 − p j ) {\displaystyle \sum \limits _{l=0}^{k}\sum \limits _{A\in F_{l}}\prod \limits _{i\in A}p_{i}\prod \limits _{j\in A^{c}}(1-p_{j})} Mean ∑ i = 1 n p i {\displaystyle \sum \limits _{i=1}^{n}p_{i}} Variance σ 2 = ∑ i = 1 n ( 1 − p i ) p i {\displaystyle \sigma ^{2}=\sum \limits _{i=1}^{n}(1-p_{i})p_{i}} |
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.
Contents
In other words, it is the probability distribution of the number of successes in a sequence of n independent yes/no experiments with success probabilities
Mean and variance
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 (
Probability mass function
The probability of having k successful trials out of a total of n can be written as the sum
where
As long as none of the success probabilities are equal to one, one can calculate the probability of k successes using the recursive formula
where
The recursive formula is not numerically stable, and should be avoided if
where
Still other methods are described in .
Entropy
There is no simple formula for the entropy of a Poisson binomial distribution, but the entropy is bounded above by the entropy of a binomial distribution with the same number parameter and the same mean. Therefore, the entropy is also bounded above by the entropy of a Poisson distribution with the same mean.
The Shepp–Olkin conjecture, due to Lawrence Shepp and Ingram Olkin in 1981, states that the entropy of a Poisson binomial distribution is a concave function of the success probabilities