# Recognize distribution moments

> ML preparation item.

- Stable ID: `original-prob-distributions`
- Area: Mathematics
- Topic: Probability
- Origin: original
- Expected time: 8 minutes
- Skills: discrete-random-variables, continuous-random-variables, named-distributions, sums-of-random-variables, expectation, variance
- Item JSON: https://mlprep.iwase.dev/data/items/original-prob-distributions.json

## Problem

Give the mean and variance of $\operatorname{Binomial}(n,p)$, $\operatorname{Poisson}(\lambda)$, $\operatorname{Exponential}(\lambda)$ under the rate parameterization, and $\mathcal{N}(\mu,\sigma^2)$. Then identify the distribution of a sum of independent Poisson variables.


## 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-distributions`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.