Binomial
The number of successes in n independent Bernoulli trials.
Probability mass
P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏdiscrete
Binomial in motion
Support{0, …, n}
Mean4.8
Variance2.88
ConnectionsWhere the binomial
Where the binomial
comes from — and leads.
Every link is one of the four moves from the relationship map: special cases, transformations, limits, and Bayesian conjugacy.
Bernoulli
← Σ iid trialsA sum of n independent Bernoulli(p) variables is Binomial(n,p).
TransformationBernoullin = 1A binomial with one trial is Bernoulli.
Special casePoissonn→∞, np=λRare independent events converge to a Poisson count.
Limiting resultNormalnp, n(1−p) largeStandardized binomial counts approach the normal law.
Limiting resultBeta← p ~ BetaBeta is conjugate to a binomial likelihood.
BayesianZoom out
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