ML Prep

Method of moments and Bayesian updating

Problem

For Exponential⁡(λ)\operatorname{Exponential}(\lambda) data under the rate parameterization, derive the method-of-moments estimate. Separately, with a Beta⁡(2,2)\operatorname{Beta}(2,2) prior and 77 successes in 1010 Bernoulli trials, give the posterior and its MAP.

Reveal answer or reference solution

Matching E[X]=1/λ\mathbb{E}[X]=1/\lambda to the sample mean gives

λ^=1X‾.\widehat{\lambda} = \frac{1}{\overline{X}}.

The posterior is Beta⁡(9,5)\operatorname{Beta}(9,5), whose MAP is

9−19+5−2=23.\frac{9-1}{9+5-2} = \frac{2}{3}.

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