# Bias, variance, MSE, and consistency

> ML Prep screening item.

- Stable ID: `math-stats-02-shrunken-proportion`
- Collection: screening
- Area: Mathematics
- Topic: Statistics
- Origin: original
- Difficulty: medium
- Expected time: 6 minutes
- Skills: estimator-bias, estimator-variance, mean-squared-error, consistency
- Item JSON: https://mlprep.iwase.dev/item/math-stats-02-shrunken-proportion.json

## Problem

For i.i.d. $X_i\sim\operatorname{Bernoulli}(p)$, let $S_n=\sum_iX_i$ and
$T_n=(S_n+1)/(n+2)$. Derive the bias, variance, and MSE of $T_n$ as an estimator
of $p$. Is $T_n$ consistent? Evaluate the bias and MSE at $n=10,p=1/2$.


## 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-stats-02-shrunken-proportion`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.