Trials until the first hit, the geometric MLE

Question

A quality-control bot keeps trying an operation until it first succeeds and records the number of attempts each time. Model each record as $X \sim \text{Geometric}(p)$ on $\{1, 2, 3, \dots\}$ , with $P(X = k) = (1-p)^{k-1} p$ . You have $n$ independent records $x_1, \dots, x_n$ .

Find the maximum likelihood estimate of the success probability $p$ .

Accepted Answer

Likelihood. Multiply the probabilities: L(p) = \prod{i=1}^n (1-p)^{xi - 1}\, p = p^{\,n}\,(1-p)^{\sumi xi - n}. Take logs: \ell(p) = n\ln p + \big(	extstyle\sumi xi - n\big)\ln(1-p). Maximize. Differentiate with respect to p: \ell'(p) = \frac{n}{p} - \frac{\sumi xi - n}{1-p} = 0. Cross-multiplying, n(1-p) = \big(\sumi xi - n\big)p, which simplifies to n = p\sumi xi, giving \hat p = \frac{n}{\sumi xi} = \frac{1}{\bar x}. This is exactly what intuition demands: a geometric variable has mean 1/p, so estimating the mean by \bar x and inverting recovers p, an instance of the MLE's invariance prop…

Trials until the first hit, the geometric MLE

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