Method of moments and Bayesian updating

Problem

For $\operatorname{Exponential}(\lambda)$ data under the rate parameterization, derive the method-of-moments estimate. Separately, with a $\operatorname{Beta}(2,2)$ prior and $7$ successes in $10$ Bernoulli trials, give the posterior and its MAP.

Reveal answer or reference solution

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

\widehat{\lambda} = \frac{1}{\overline{X}}.

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

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

Local history

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Statistics Practice