chapter2.qmd

# Chapter 2: Advanced Topics

This is the second chapter. Continue building your book with more chapters.

## Mathematical Equations

You can include mathematical equations using LaTeX syntax:

Inline equation: $E = mc^2$

Display equation:

$$
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
$$

## Custom Macros

This book uses the [`d-morrison/macros`](https://github.com/d-morrison/macros) submodule to provide convenient LaTeX shorthand for statistical and mathematical notation.

### Probability and Distributions

The normal distribution $\Normal\paren{\m, \ss}$ can be written using macros as `$\Normal\paren{\m, \ss}$`, where `\m` expands to $\mu$ and `\ss` expands to $\sigma^2$.

A random variable $X \dist \Normal\paren{\mu, \sigma^2}$ has expectation $\E{X} = \mu$ and variance $\Var{X} = \sigma^2$.

### Regression Notation

In linear regression, we estimate $\vbeta$ using ordinary least squares. The fitted values are $\hy = \mX \hb$, where $\hb = \inv{\mX' \mX} \mX' \vy$.

The standard error of $\hb$ is $\hse{\hb} = \hs \sqrt{\inv{\mX' \mX}}$.

### Likelihood and Estimation

The log-likelihood function $\llik(\th)$ is maximized at the MLE $\hth_{\text{ML}}$.

The score function is $\score(\th) = \deriv{\th} \llik(\th)$ and the observed information is $\oinf(\th) = -\hess(\th)$.

### Logistic Regression

The logit link function is $\logitf{\pi} = \logt\paren{\frac{\pi}{1 - \pi}}$ and its inverse is $\expitf{\eta} = \frac{e^\eta}{1 + e^\eta}$.

## Tables

| Column 1 | Column 2 | Column 3 |
|----------|----------|----------|
| Data 1   | Data 2   | Data 3   |
| Data 4   | Data 5   | Data 6   |

: Example table caption {#tbl-example}

## Figures

You can include images by placing them in the `images/` directory:

```markdown
![Example image caption](images/your-image.png){#fig-example}
```

Then reference the figure in text using `@fig-example`.

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