Binomiallikelihood

Binomiallikelihood, or the binomial likelihood, is the likelihood function for a binomial distribution. It describes the probability of observing a given number of successes in a fixed number of independent Bernoulli trials as a function of the success probability p.

If Y follows a Binomial(n, p) distribution and y is the observed number of successes, the likelihood of p is L(p) = C(n, y) p^y (1-p)^(n−y). The binomial coefficient C(n, y) does not depend on p, so the likelihood (as a function of p) is proportional to p^y (1-p)^(n−y) for p in [0, 1].

Maximum likelihood estimation using the binomial likelihood yields the estimator p̂ = y/n, the observed proportion of successes. The log-likelihood is l(p) = y log p + (n − y) log(1 − p) + constant, and its maximization confirms p̂ as the solution (interior unless y = 0 or y = n).

Properties of the binomial likelihood include its domain p ∈ [0, 1] and its unimodal shape for typical data. Its curvature is described by the Fisher information I(p) = n/(p(1 − p)), which governs the precision of estimates.

Applications and extensions: the binomial likelihood is foundational for frequentist inference, including hypothesis testing and likelihood-based confidence intervals, and for Bayesian analysis with conjugate Beta priors, which yield Beta posterior distributions. The likelihood approach emphasizes the data’s support for different p values without requiring priors or sampling distributions beyond the binomial model.