# Method of moments and Bayesian updating

> ML preparation item.

- Stable ID: `original-stats-mom-map`
- Area: Mathematics
- Topic: Statistics
- Origin: original
- Expected time: 10 minutes
- Skills: method-of-moments, map, bayesian-updating, likelihood
- Item JSON: https://mlprep.iwase.dev/data/items/original-stats-mom-map.json

## 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.


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Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `original-stats-mom-map`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.