# LLN versus CLT

> ML preparation item.

- Stable ID: `original-prob-lln`
- Area: Mathematics
- Topic: Probability
- Origin: original
- Expected time: 5 minutes
- Skills: law-of-large-numbers, central-limit-theorem
- Item JSON: https://mlprep.iwase.dev/data/items/original-prob-lln.json

## Problem

For i.i.d. variables with mean $\mu$ and finite variance $\sigma^2$, state what the law of large numbers says about the sample mean $\overline{X}_n$ and what the central limit theorem says about its standardized error.


## Answer policy

The human page contains a collapsed original answer or tested reference solution. 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 `original-prob-lln`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.