# GMM responsibilities and M-step

> ML Prep screening item.

- Stable ID: `ml-screen-unsupervised-standard-gmm`
- Collection: screening
- Area: Machine Learning
- Topic: Unsupervised Learning
- Origin: original
- Difficulty: medium
- Expected time: 9 minutes
- Skills: gaussian-mixtures, expectation-maximization
- Item JSON: https://mlprep.iwase.dev/item/ml-screen-unsupervised-standard-gmm.json

## Problem

Two GMM components have weights $(0.25,0.75)$ and densities at $x$ of $(0.6,0.2)$. Compute the responsibilities. Then, observations $(0,10)$ have responsibility rows $(0.5,0.5)$ and $(0.8,0.2)$. Compute the updated weights and means.


## 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 `ml-screen-unsupervised-standard-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.