Bernoulli Distribution
X ~ Bernoulli(p)
Single-Trial Family • Published July 20, 2026
The single yes/no trial underlying the entire success/failure family on this site: history from Jacob Bernoulli's posthumous 1713 Ars Conjectandi, its exact identity as the n=1 case of the already-published Binomial, why a fair coin (p=0.5) simultaneously maximizes variance and entropy, a fully closed-form maximum-likelihood estimator that exactly attains the Cramér-Rao bound, and direct simulation.
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About This Reference Sheet
The single yes/no trial underlying the entire success/failure family on this site: history from Jacob Bernoulli's posthumous 1713 Ars Conjectandi, its exact identity as the n=1 case of the already-published Binomial, why a fair coin (p=0.5) simultaneously maximizes variance and entropy, a fully closed-form maximum-likelihood estimator that exactly attains the Cramér-Rao bound, and direct simulation.
Support
x ∈ {0, 1}
Parameters
p ∈ [0,1] (success probability)