---
title: "Biostat 212A Homework 3"
subtitle: "Due Feb 22, 2026 @ 11:59PM"
author: "YOUR NAME and UID"
date: today
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## ISL Exercise 5.4.2 (10pts)

## ISL Exercise 5.4.9 (20pts)

## Least squares is MLE (10pts)

Show that in the case of linear model with Gaussian errors, maximum likelihood and least squares are the same thing, and $C_p$ and AIC are equivalent.

## ISL Exercise 6.6.1 (10pts)

## ISL Exercise 6.6.3 (10pts)

## ISL Exercise 6.6.4 (10pts)

## ISL Exercise 6.6.5 (10pts)

## ISL Exercise 6.6.11 (30pts)

You must follow the [typical machine learning paradigm](https://ucla-biostat-212a.github.io/2024winter/slides/06-modelselection/workflow_lasso.html) to compare _at least_ 3 methods: least squares, lasso, and ridge. Report final results as

| Method | CV RMSE | Test RMSE |
|:------:|:------:|:------:|:------:|
| LS | | | |
| Ridge | | | |
| Lasso | | | |
| ... | | | |

## Bonus question (20pts)

Consider a linear regression, fit by least squares to a set of training data $(x_1, y_1), \ldots, (x_N,  y_N)$ drawn at random from a population. Let $\hat \beta$ be the least squares estimate. Suppose we have some test data $(\tilde{x}_1, \tilde{y}_1), \ldots, (\tilde{x}_M, \tilde{y}_M)$ drawn at random from the same population as the training data. If $R_{\text{train}}(\beta) = \frac{1}{N} \sum_{i=1}^N (y_i - \beta^T x_i)^2$ and $R_{\text{test}}(\beta) = \frac{1}{M} \sum_{i=1}^M (\tilde{y}_i - \beta^T \tilde{x}_i)^2$. Show that
$$
\operatorname{E}[R_{\text{train}}(\hat{\beta})] < \operatorname{E}[R_{\text{test}}(\hat{\beta})].
$$