Poisson binomial distribution

Poisson binomial

Parameters	${\displaystyle \mathbf {p} \in [0,1]^{n}}$ — success probabilities for each of the n trials
Support	k ∈ { 0, …, n }
PMF	${\displaystyle \sum \limits _{A\in F_{k}}\prod \limits _{i\in A}p_{i}\prod \limits _{j\in A^{c}}(1-p_{j})}$
CDF	${\displaystyle \sum \limits _{l=0}^{k}\sum \limits _{A\in F_{l}}\prod \limits _{i\in A}p_{i}\prod \limits _{j\in A^{c}}(1-p_{j})}$
Mean	${\displaystyle \sum \limits _{i=1}^{n}p_{i}}$
Variance	${\displaystyle \sigma ^{2}=\sum \limits _{i=1}^{n}(1-p_{i})p_{i}}$
Skewness	${\displaystyle {\frac {1}{\sigma ^{3}}}\sum \limits _{i=1}^{n}(1-2p_{i})(1-p_{i})p_{i}}$
Excess kurtosis	${\displaystyle {\frac {1}{\sigma ^{4}}}\sum \limits _{i=1}^{n}(1-6(1-p_{i})p_{i})(1-p_{i})p_{i}}$
MGF	${\displaystyle \prod \limits _{j=1}^{n}(1-p_{j}+p_{j}e^{t})}$
CF	${\displaystyle \prod \limits _{j=1}^{n}(1-p_{j}+p_{j}e^{it})}$
PGF	${\displaystyle \prod \limits _{j=1}^{n}(1-p_{j}+p_{j}z)}$

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 ${\displaystyle p_{1},p_{2},\dots ,p_{n}}$. The ordinary binomial distribution is a special case of the Poisson binomial distribution, when all success probabilities are the same, that is ${\displaystyle p_{1}=p_{2}=\cdots =p_{n}}$.