# Build and maximize a likelihood

> ML Prep screening item.

- Stable ID: `math-stats-03-exponential-mle`
- Collection: screening
- Area: Mathematics
- Topic: Statistics
- Origin: original
- Difficulty: medium
- Expected time: 4 minutes
- Skills: likelihood, maximum-likelihood
- Item JSON: https://mlprep.iwase.dev/item/math-stats-03-exponential-mle.json

## Problem

Observations $1,2,3$ are i.i.d. from an exponential distribution with unknown
rate $\lambda>0$. Write the likelihood and log-likelihood, then find the MLE.


## Answer policy

The human page contains a collapsed answer. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `math-stats-03-exponential-mle`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.