Appendix C: Mathematical Notation Summary - Learn Probability

This appendix provides a summary of the common mathematical notations used throughout this book. Familiarity with these symbols is helpful for understanding the theoretical underpinnings alongside the Python implementations.

Set Theory and Probability Basics¶

Notation	Meaning	Example	Chapter(s)
$S$ , $\Omega$	Sample Space (the set of all possible outcomes)	$S = \{1, 2, 3, 4, 5, 6\}$ for a die roll.	2
$A, B, E, ...$	Events (subsets of the sample space)	$A = \{2, 4, 6\}$ (rolling an even number).	2
$\emptyset$	Empty Set (impossible event)	Rolling a 7 on a standard die.	2
$A \cup B$	Union (‘A or B’ or both occur)	$\{1, 2, 3\} \cup \{3, 4, 5\} = \{1, 2, 3, 4, 5\}$	2
$A \cap B$	Intersection (‘A and B’ both occur)	$\{1, 2, 3\} \cap \{3, 4, 5\} = \{3\}$	2
$A^c$ , $\bar{A}$	Complement (‘not A’)	If $S=\{1,2,3\}$ , $A=\{1\}$ , then $A^c = \{2, 3\}$ .	2
$A \setminus B$	Set Difference (‘A but not B’)	$\{1, 2, 3\} \setminus \{3, 4, 5\} = \{1, 2\}$	2
$	A	$	Cardinality (number of elements in set A)
$P(A)$	Probability of event A occurring	$P(\text{Heads}) = 0.5$ for a fair coin.	2
$P(A	B)$	Conditional Probability (prob. of A given B)	$P(\text{Sum}>10

Counting Techniques¶

Notation	Meaning	Example	Chapter(s)
$n!$	Factorial ( $n \times (n-1) \times ... \times 1$ )	$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$	3
$P(n, k)$ , $^nP_k$	Permutations (ordered arrangements of k from n)	Ways to award Gold, Silver, Bronze to 3 of 10 runners	3
$C(n, k)$ , $^nC_k$ , $\binom{n}{k}$	Combinations (unordered selections of k from n)	Ways to choose a committee of 3 from 10 people	3

Random Variables and Distributions¶

Notation	Meaning	Example	Chapter(s)
$X, Y, Z$	Random Variables (variables whose values are numerical outcomes)	$X =$ Number of heads in 3 coin flips.	6-12
$x, y, z$	Specific values (realizations) of random variables	$X$ could take the value $x=2$ .	6-12
$X \sim \text{Dist}(...)$	‘X follows the distribution Dist with given parameters’	$X \sim \text{Binomial}(n=10, p=0.5)$	7, 9
$p(x)$ , $p_X(x)$ , $P(X=x)$	Probability Mass Function (PMF) of a discrete RV $X$	$p_X(k) = P(X=k)$ for $k=0, 1, ..., n$ in a Binomial distribution.	6, 7
$f(x)$ , $f_X(x)$	Probability Density Function (PDF) of a continuous RV $X$	The bell curve shape for a Normal distribution.	8, 9
$F(x)$ , $F_X(x)$	Cumulative Distribution Function (CDF) $P(X \le x)$	$F_X(x) = P(X \le x)$	6, 8
$E[X]$ , $\mu$ , $\mu_X$	Expected Value (mean) of RV $X$	Average value expected from many trials.	6, 8
$Var(X)$ , $\sigma^2$ , $\sigma^2_X$	Variance of RV $X$ (measure of spread)	$Var(X) = E[(X - \mu)^2]$	6, 8
$SD(X)$ , $\sigma$ , $\sigma_X$	Standard Deviation of RV $X$ ( $\sqrt{Var(X)}$ )	Spread measured in the same units as $X$ .	6, 8

Multiple Random Variables¶

Notation	Meaning	Chapter(s)
$(X, Y)$	A pair of random variables	10-12
$p(x, y)$ , $p_{X,Y}(x, y)$	Joint PMF of discrete RVs $X, Y$	10
$f(x, y)$ , $f_{X,Y}(x, y)$	Joint PDF of continuous RVs $X, Y$	10
$F(x, y)$ , $F_{X,Y}(x, y)$	Joint CDF $P(X \le x, Y \le y)$	10
$p_X(x)$ , $f_X(x)$	Marginal PMF/PDF of $X$ (derived from joint distribution)	10
$p(y	x) $,$ p_{Y	X}(y
$f(y	x) $,$ f_{Y	X}(y
$Cov(X, Y)$	Covariance between $X$ and $Y$ ( $E[(X-\mu_X)(Y-\mu_Y)]$ )	11
$\rho(X, Y)$ , $Corr(X, Y)$	Correlation Coefficient between $X$ and $Y$ ( $\frac{Cov(X,Y)}{\sigma_X \sigma_Y}$ )	11

Limit Theorems and Convergence¶

Notation	Meaning	Chapter(s)
$X_n \xrightarrow{p} X$	Convergence in Probability	13
$X_n \xrightarrow{d} X$	Convergence in Distribution	14

Bayesian Inference¶

Notation	Meaning	Chapter(s)
$\theta$	Parameter of interest	5, 15
$\pi(\theta)$	Prior distribution of $\theta$	15
$L(\theta	x)$	Likelihood function
$p(\theta	x)$	Posterior distribution of $\theta$

Markov Chains¶

Notation	Meaning	Chapter(s)
$P_{ij}$	Transition probability from state $i$ to $j$	16
$\mathbf{P}$	Transition Probability Matrix	16
$\pi$	Stationary distribution vector	16

General Mathematical Symbols¶

Notation	Meaning	Chapter(s)
$\sum$	Summation	Throughout
$\int$	Integral	Throughout
$\approx$	Approximately equal to	Throughout
$\propto$	Proportional to	5, 15
$\mathbb{R}$	Set of real numbers	Throughout
$\mathbb{N}$	Set of natural numbers (usually $\{1, 2, 3, ...\}$ )	Throughout
$\in$	‘Element of’ or ‘belongs to’	2
$\forall$	‘For all’	Throughout
$\exists$	‘There exists’	Throughout