PSTAT 120B: Mathematical Statistics, I

Statistical Notation

Instructor

Quarter

Ethan Marzban

Summer Session A, 2024

I am sympathetic to the fact that, at times, the notation used in statistics can seem a bit overwhelming. As such, I’ve crafted this page to outline a few key notations and symbols used in Statistics - I encourage you to refer to this page often! (Of course, please feel free to ask me directly in Lecture or Office Hours if I use notation that is unfamiliar to you.)

Greek Alphabet

Table 1: The Greek Alphabet

First Half

Uppercase	Lowercase	Name
A	$\alpha$	alpha
B	$\beta$	beta
$\mathrm{\Gamma }$	$\gamma$	gamma
$\mathrm{\Delta }$	$\delta$	delta
E	$\epsilon$	epsilon
Z	$\zeta$	zeta
H	$\eta$	nu
$\mathrm{\Theta }$	$\theta$ / $\vartheta$	theta
I	$\iota$	iota
K	$\kappa$	kappa
$\mathrm{\Lambda }$	$\lambda$	lambda
M	$\mu$	mu

Second Half

Uppercase	Lowercase	Name
N	$\nu$	nu
$\mathrm{\Xi }$	$\xi$	xi
O	$o$	omicron
$\mathrm{\Pi }$	$\pi$	pi
P	$\rho$	rho
$\mathrm{\Sigma }$	$\sigma$ / $\varsigma$	sigma
T	$\tau$	tau
Y	$\upsilon$	upsilon
$\mathrm{\Phi }$	$\varphi$	phi
X	$\chi$	chi
$\mathrm{\Psi }$	$\psi$	psi
$\mathrm{\Omega }$	$\omega$	omega

Some Common Statistical Uses

• $\mu$: often used to denote mean/expected value
• $\sigma$: often used to denote standard deviations
• $\theta$: often used as a placeholder for an arbitrary parameter
• $\lambda ,\rho$: often used to denote rates
• $\nu$: often used to denote the degrees of freedom of a ${\chi }^2$ distribution
• $\mathrm{\Gamma }(t)$: the Gamma function $$\mathrm{\Gamma }(r):={\int }_0^{\mathrm{\infty }}{x}^{r-1}{e}^{-x}\mathrm{d}x$$
• $\psi (t)$: the Digamma function: $$\psi (t):=\frac{\mathrm{d}}{\mathrm{d}t}\ln [\mathrm{\Gamma }(t)]=\frac{{\mathrm{\Gamma }}^{\prime }(t)}{\mathrm{\Gamma }(t)}$$
• Sometimes, related parameters will be named after consecutive letters in the alphabet. For instance, we may denote the mean of a random variable $X$ by $\mu$ and the mean of another random variable $Y$ by $\nu$, and $\mu$ and $\nu$ appear consecutively in the Greek alphabet. Similarly for variances; ${\sigma }^2$ and ${\tau }^2$ are commonly used to denote variances.

Other Notations

• In this class, I will often write $X\sim f_X$ to mean “let $X$ be a random variable with density function $f_X(x)$”, and $X\sim F_X$ to mean “let $X$ be a random variable with distribution function $F_X(x)$.”

• The symbol $\mathrm{\forall }$ is called the universal quantifier, and is essentially a mathematical way of writing “for all” or “for every”. As an example: $$(\mathrm{\forall }x\in \mathbb{R})[f_X(x)\ge 0]$$ is read “for every real number $x$, the value of $f_X(x)$ will be nonnegative”.

• The symbol $\mathrm{\exists }$ is called the existential quantifier, and is essentially a mathematical way of writing “there exists” or “for at least one”. As an example: $$(\mathrm{\exists }x\in \mathbb{R})[x^2=2]$$ is read “there exists a real number $x$ whose square is 2”.

• I often use the symbol $:=$ to mean “definitionally equal to”. For example, given a random variable $Y$, I’ll often write $U:=Y^2$ to mean “let $U$ be another random variable that is, by definition, equal to the square of $Y$.”

• I often use the symbol $\stackrel{!}{=}$ to mean “set equal to”. For example, solving $f^{\prime }(x)\stackrel{!}{=}0$ for $x$ gives us the set of critical values for the function $f(\cdot )$.