Mathematical Notation

A compact shared vocabulary of symbols used throughout math and statistics. Knowing them lets you read likelihoods, probability statements, and model definitions without stumbling.

Sets and logic#

We often work with the real numbers $\mathbb{R}$ . A set $A$ can be a subset of another, $A \subset B$ , and an object can be an element of a set, $x \in A$ . Two events combine via union $A \cup B$ (either) and intersection $A \cap B$ (both).

Logical shorthand: for all $\forall$ , there exists $\exists$ , and implies $\Rightarrow$ . In probability we write distributed as $X \sim \mathcal{N}(0,1)$ , and independence as $X \perp Y$ (or the statistical form $X \perp\!\!\!\perp Y$ ).

Sums, products, and counting#

The summation $\sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n$ adds terms; the product $\prod_{i=1}^{n} a_i = a_1 a_2 \cdots a_n$ multiplies them. The factorial is $n! = n\,(n-1)\cdots 2 \cdot 1$ , and the binomial coefficient counts subsets:

$\binom{n}{k} = \frac{n!}{k!\,(n-k)!}.$

Symbol reference#

Symbol	Meaning	LaTeX
$\mathbb{R}$	real numbers	`\mathbb{R}`
$\in$	element of	`\in`
$\subset$	subset of	`\subset`
$\cup$	union	`\cup`
$\cap$	intersection	`\cap`
$\forall$	for all	`\forall`
$\exists$	there exists	`\exists`
$\Rightarrow$	implies	`\Rightarrow`
$\sim$	distributed as	`\sim`
$\perp\!\!\!\perp$	independent	`\perp\!\!\!\perp`
$\sum$	summation	`\sum`
$\prod$	product	`\prod`
$n!$	factorial	`n!`
$\binom{n}{k}$	binomial coefficient	`\binom{n}{k}`

Writing formulas in LaTeX#

Fractions: \frac{a}{b} renders as $\frac{a}{b}$ .
Greek letters: \alpha, \beta, \mu, \sigma render as $\alpha, \beta, \mu, \sigma$ .
Sums and products: \sum_{i=1}^{n} i gives $\sum_{i=1}^{n} i$ and \prod_{i=1}^{n} i gives $\prod_{i=1}^{n} i$ .
Distributed as: X \sim \mathcal{N}(\mu, \sigma^2) gives $X \sim \mathcal{N}(\mu, \sigma^2)$ .

Worked example#

For $n = 5$ and $k = 2$ :

$\sum_{i=1}^{5} i = 15,\qquad \prod_{i=1}^{5} i = 5! = 120,\qquad \binom{5}{2} = \frac{120}{2 \cdot 6} = 10.$

Computing it#

R#

n <- 5; k <- 2
sum(1:n)        # 15
prod(1:n)       # 120
factorial(n)    # 120
choose(n, k)    # 10

Python#

Python

import math
n, k = 5, 2
print(sum(range(1, n + 1)))       # 15
print(math.prod(range(1, n + 1))) # 120
print(math.factorial(n))          # 120
print(math.comb(n, k))            # 10

Julia#

Julia

n, k = 5, 2
sum(1:n)          # 15
prod(1:n)         # 120
factorial(n)      # 120
binomial(n, k)    # 10

Why it matters for statistics#

Statistical models are written in this notation: likelihoods are products $\prod_i f(x_i)$ , expectations are sums $\sum_i x_i p_i$ , and independence assumptions ( $X \perp\!\!\!\perp Y$ ) justify factoring joint distributions. Fluency here is the prerequisite for everything that follows.