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 the number of successes in a collection 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}} .
Definitions
Probability mass function The probability of having k successful trials out of a total of n can be written as the sum
Pr ( K = k ) = ∑ A ∈ F k ∏ i ∈ A p i ∏ j ∈ A c ( 1 − p j ) {\displaystyle \Pr(K=k)=\sum \limits _{A\in F_{k}}\prod \limits _{i\in A}p_{i}\prod \limits _{j\in A^{c}}(1-p_{j})}
where F k {\displaystyle F_{k}} is the set of all subsets of k integers that can be selected from { 1 , 2 , 3 , . . . , n } {\displaystyle \{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\}\right\}} . 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\}\smallsetminus 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
Pr ( K = k ) = { ∏ i = 1 n ( 1 − p i ) k = 0 1 k ∑ i = 1 k ( − 1 ) i − 1 Pr ( K = k − i ) T ( i ) k > 0 {\displaystyle \Pr(K=k)={\begin{cases}\prod \limits _{i=1}^{n}(1-p_{i})&k=0\\{\frac {1}{k}}\sum \limits _{i=1}^{k}(-1)^{i-1}\Pr(K=k-i)T(i)&k>0\\\end{cases}}}
where
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