# Transform an exponential variable

> ML Prep screening item.

- Stable ID: `math-prob-04-exponential-transform`
- Collection: screening
- Area: Mathematics
- Topic: Probability
- Origin: original
- Difficulty: medium
- Expected time: 4 minutes
- Skills: continuous-random-variables, pdf, cdf, transformations, named-distributions
- Item JSON: https://mlprep.iwase.dev/item/math-prob-04-exponential-transform.json

## Problem

Let $X\sim\operatorname{Exponential}(2)$ under the rate parameterization and
define $Y=2X$. Derive the CDF and PDF of $Y$, identify its distribution, and
give its mean and variance.


## 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-prob-04-exponential-transform`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.