1. Notation Reference

Variables and Values

Symbol	Meaning
$X,Y,Z$	Random variables (capital letters)
$x,y,z$	Specific realized values of the corresponding random variables
$\mathcal{X},\mathcal{Y}$	Alphabets or supports of random variables; for example, $x\in \mathcal{X}$
$\mathbf{x},\mathbf{y}$	Vectors (bold lowercase); e.g. $\mathbf{x}\in {\mathbb{R}}^d$
$\mathbf{X}$	A matrix (bold uppercase); e.g. design matrix $\mathbf{X}\in {\mathbb{R}}^{N\times d}$

Probability

Symbol	Meaning
$P_X$	Probability distribution, or law, of the random variable $X$
$p_X(x)=\mathrm{Pr}(X=x)$	PMF of a discrete random variable $X$, evaluated at the realization $x$
$f_X(x)$	PDF of a continuous random variable $X$, evaluated at $x$
$p_{X,Y}(x,y)$	Joint PMF of discrete random variables $(X,Y)$, evaluated at $(x,y)$
$p_{Y\mid X}(y\mid x)$	Conditional PMF of $Y=y$ given the event $X=x$
$P_{Y\mid X}$	Conditional distribution of the random variable $Y$ given $X$
$p_{\mathbf{X}}(\mathbf{x})$	Joint PMF/PDF of a random vector $\mathbf{X}$, evaluated at $\mathbf{x}$
$p_{\theta }(\mathbf{x})$	Parametric PMF/PDF evaluated at $\mathbf{x}$, with parameter $\theta$
$p_{\theta }(\mathbf{x}\mid \mathbf{y})$	Parametric conditional PMF/PDF evaluated at $(\mathbf{x},\mathbf{y})$

When the associated random variables are obvious, subscripts may be suppressed for readability: $p_X(x)$ becomes $p(x)$ and $p_{Y\mid X}(y\mid x)$ becomes $p(y\mid x)$. Uppercase arguments such as $p_X(X)$ denote the PMF evaluated at the random outcome $X$ and therefore produce a random variable.

Datasets and Expectations

Symbol	Meaning
$\mathcal{D}=\{{\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(N)}\}$	Dataset containing $N$ realized samples
${\mathbf{X}}^{(i)}\stackrel{i.i.d.}{\sim }{P}_{\mathbf{X}}$	Random samples drawn independently from the same distribution
${\mathbb{E}}_{X\sim P_X}[g(X)]$	Expectation of the random variable $g(X)$ under $P_X$
${\mathbb{E}}_{x\sim p_{\mathrm{data}}}[g(x)]$	Common machine-learning shorthand for an expectation over data realizations

Common Distributions

Symbol	Meaning
$\mathcal{N}(\mu ,{\sigma }^2)$	Univariate Gaussian with mean $\mu$ and variance ${\sigma }^2$
$\mathcal{N}(\mathbf{\mu },\mathbf{\Sigma })$	Multivariate Gaussian with mean $\mathbf{\mu }$ and covariance $\mathbf{\Sigma }$
$\mathcal{N}(0,\mathbf{I})$	Standard multivariate Gaussian (zero mean, identity covariance)
$\text{Bernoulli}(p)$	Bernoulli distribution with success probability $p$