Binomial Distribution
X ~ Binomial(n, p)
Sum-of-Bernoulli-Trials Family • Published July 18, 2026
The count of successes in n independent trials: history from Bernoulli through de Moivre's normal approximation, the PMF and why discrete hazard/quantile functions are defined differently than in the continuous case, the full property list, maximum-likelihood estimation of p (exactly unbiased, exactly efficient), and simulation as a direct sum of Bernoulli draws.
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About This Reference Sheet
The count of successes in n independent trials: history from Bernoulli through de Moivre's normal approximation, the PMF and why discrete hazard/quantile functions are defined differently than in the continuous case, the full property list, maximum-likelihood estimation of p (exactly unbiased, exactly efficient), and simulation as a direct sum of Bernoulli draws.
Support
x ∈ {0, 1, 2, ..., n}
Parameters
n ∈ ℕ (trials), p ∈ [0,1] (success probability)