= Poisson binomial distribution
{c}
{title2=$K=\sum_{i=1}^nX_i,\quad X_i\sim\operatorname{Bernoulli}(p_i)$}
{wiki}
The sum of <independent> <Bernoulli random variables> with possibly unequal success <probabilities> $p_i$ has a Poisson binomial distribution. Its <probability generating function> is $G_K(z)=\prod_i(1-p_i+p_i z)$: expand the product to read off the <probability mass function>. <Linearity of expectation> and additivity of <variance> for <independent random variables> give $\mathbb EK=\sum_i p_i$ and $\operatorname{Var}(K)=\sum_i p_i(1-p_i)$. Equal $p_i=p$ recover the <binomial distribution>.
For three trials with success <probabilities> $1/10,3/10,4/5$, expanding $(9/10+z/10)(7/10+3z/10)(1/5+4z/5)$ gives masses $0.126,0.572,0.278,0.024$ at $0,1,2,3$. Their <expected value> is $1.2$ and <variance> is $0.46$. Replacing all three success <probabilities> by their average $0.4$ preserves the <expected value> but gives <binomial distribution> <variance> $0.72$, so it does not preserve the law.
Back to article page
