Appendix C: Mathematical Notation Summary#
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 |