# Mixture responsibilities and EM

> ML preparation item.

- Stable ID: `original-ml-gmm`
- Area: Machine Learning
- Topic: Unsupervised Learning
- Origin: original
- Expected time: 7 minutes
- Skills: gaussian-mixtures, expectation-maximization, clustering-objective
- Item JSON: https://mlprep.iwase.dev/data/items/original-ml-gmm.json

## Problem

At a data point $x$, two mixture components have prior weights $0.4$ and $0.6$ and likelihoods $p(x\mid z=1)=0.3$ and $p(x\mid z=2)=0.1$. Compute posterior responsibilities and state the E- and M-step roles.


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