Course objectives and notation

Probability provides a mathematical language for uncertainty. A probability model begins with a random experiment, identifies the possible outcomes, groups outcomes into events and assigns probabilities consistently. The same framework supports applications in quality control, engineering reliability, finance, public health, agriculture, telecommunications and scientific research.

The course is organised into three broad sections:

• Counting and probability laws: Lectures One to Four.
• Random variables and discrete distributions: Lectures Five to Ten.
• Continuous, joint and transformed distributions: Lectures Eleven to Fifteen.

Course Objectives

By the end of the course, students should be able to:

1. construct sample spaces and events for random experiments;
2. apply permutations, combinations and multinomial counting methods;
3. use the axioms and laws of probability, including conditional probability and Bayes’ theorem;
4. define random variables and calculate expectations, variances and moments;
5. identify and apply standard discrete and continuous probability distributions;
6. derive marginal and conditional distributions from joint distributions;
7. transform random variables using distribution and Jacobian methods; and
8. explain the probabilistic foundations of the Law of Large Numbers and the Central Limit Theorem.

Core Notation

Symbol	Meaning
$\Omega$	sample space
$A,B,C$	events
$A^{c}$	complement of $A$
$P(A)$	probability of event $A$
$P(A \mid B)$	conditional probability of $A$ given $B$
$X,Y$	random variables
$p_{X}(x)$	probability mass function of a discrete random variable
$f_{X}(x)$	probability density function of a continuous random variable
$F_{X}(x)$	cumulative distribution function
$E(X)$	expected value of $X$
$Var(X)$	variance of $X$
$M_{X}(t)$	moment generating function of $X$