Poisson-Binomial Distribution
X ~ PoiBin(p_1, ..., p_n)
Sum-of-Independent-Bernoullis Family • Published July 22, 2026
The already-published Binomial distribution's heterogeneous generalization: a sum of independent Bernoulli trials that no longer share one probability, the exact distribution behind vote-count forecasting, portfolio credit risk, and reliability engineering, computed via an O(n²) dynamic-programming convolution verified against brute-force enumeration to machine precision, a proof that heterogeneous p_i always concentrates the sum more than a same-mean Binomial (Jensen's inequality on p(1-p)), a worked explanation of why individual p_i are not identifiable from repeated sums even though the aggregate mean and variance are, and simulation as a direct sum of independent Bernoulli draws.
This reference sheet is part of an Analyze subscription.
Subscribe to read the full Poisson-Binomial Distribution reference sheet, along with every other resource in the library.
See plansPrefer a saved copy?
About This Reference Sheet
The already-published Binomial distribution's heterogeneous generalization: a sum of independent Bernoulli trials that no longer share one probability, the exact distribution behind vote-count forecasting, portfolio credit risk, and reliability engineering, computed via an O(n²) dynamic-programming convolution verified against brute-force enumeration to machine precision, a proof that heterogeneous p_i always concentrates the sum more than a same-mean Binomial (Jensen's inequality on p(1-p)), a worked explanation of why individual p_i are not identifiable from repeated sums even though the aggregate mean and variance are, and simulation as a direct sum of independent Bernoulli draws.
Support
x ∈ {0, 1, ..., n}
Parameters
n ∈ {1,2,...} (number of trials), p_1,...,p_n ∈ [0,1] (individual success probabilities)