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

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

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.