notation.Rmd

---
title: "notation"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{notation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```


## Notational conventions in this class

-   Capital letters ("$X$"): random variables

-   Lowercase letters ("$x$"): observed data

-   Greek letters ("$\alpha$"): parameters

-   $P(X=x)$: Probability mass

    -   shorthand: $P(x) \equiv P(X=x)$

-   $p(X=x)$: Probability density

    -   shorthand: $p(x) = p(X=x)$.

-   Note: $p(x) = \frac{d}{dx} P(x)$

-  $E[X]$ \equiv $\mu[X]$ is the mean of the distribution of $X$.

More precisely, $E[X]$ is the mean of set of the possible values ("range") of $X$, weighted by their probability density or mass.

$$E[X] = \sum_{x\in R(X)} x \cdot P(X=x)$$

$$E[X] = \int_{x\in R(X)} x \cdot P(X=x) dx$$