MLE for a Bernoulli probability

Question

You observe n independent flips of a coin with unknown head probability p, and see k heads. Derive the maximum likelihood estimator of p, verify it is a maximum, and state its bias and variance.

Accepted Answer

Likelihood. Each flip contributes p (head) or 1-p (tail), so for a sequence with k heads out of n, L(p) = p^k (1-p)^{n-k}, \qquad \ell(p) = \log L(p) = k\log p + (n-k)\log(1-p). First-order condition. \ell'(p) = \frac{k}{p} - \frac{n-k}{1-p} = 0 \implies k(1-p) = (n-k)p \implies \hat{p} = \frac{k}{n}. Second-order check. \ell''(p) = -\frac{k}{p^2} - \frac{n-k}{(1-p)^2} < 0 \quad 	ext{for } 0 < p < 1, so \ell is concave and \hat{p} = k/n is the global maximum. (Edge cases: if k = 0 or k = n the MLE sits on the boundary, \hat{p} = 0 or 1, a reminder that MLEs can be degenerate in small samples…

MLE for a Bernoulli probability

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