# Notation

Conventions used throughout the book include:

- Boldface is used to denote vectors and matrices, the former in lowercase and the latter in uppercase. So $\mathbf{X}$ is a matrix and $\mathbf{x}$ is a (column) vector.

- $x$, $y$, and $z$ are generally variables, while $f$ and $g$ are functions. $X$ is typically a random variable. Other Latin and Greek letters are usually parameters or constants. 

Additional symbols include:

| **Symbol** | **Meaning** |
|-|-|
| $:=$ | Is defined as |
| $\{1, 2, \ldots, n \}$ | The set of integers from 1 to $n$. In general $\{\cdots\}$ denotes a set. |
| $x\in\mathcal{X}$ | $x$ is a member of the set $\mathcal{X}$ |
| $\displaystyle \sum_{i=1}^n x_i$ | Sum of $x_1 + \ldots + x_n$. Also written as $\sum_{i=1}^n x_i$. Alternatively, can write the summation of all elements in a set $\mathcal{X}$ as $\displaystyle \sum_{x\in \mathcal{X}} x$ or $\sum_{x\in \mathcal{X}} x$ |
| $\R$ | The real numbers |
| $\R^+$ | The positive real numbers, $\{x\in \R: x>0\}$ |
| $\forall i$ | For all $i$ |
| $\mathbf{x}'$ | The transpose of $\mathbf{x}$. Also applies to matrices. |
| $f(x) : \R \rightarrow [0,1]$ | The function $f$ maps from the real numbers to the closed interval 0 to 1 |
| $f'(x)$ or $\frac{df}{dx}$ | The derivative of $f$ with respect to $x$ |
| $\P$ | Probability measure. Maps from the space of outcomes $\Omega$ to a probability in $[0,1]$ |
| $\E(X)$ | Expectation of random variable $X$ |
| $\N(\mu,\sigma^2)$ | Normal distribution with mean $\mu$ and variance $\sigma^2$ |