AX26-STAT-101 · 2026-27
Probability
Build probability from spaces and random variables through expectation to elementary limit theorems.
- Estimated study
- 150 hours
- Level
- Undergraduate level
- Topics
- 8
Course topics
- 01Probability spaces
- 02Conditional probability and independence
- 03Discrete random variables
- 04Continuous random variables
- 05Expectation and variance
- 06Joint distributions
- 07Limit theorems
- 08Synthesis
Detailed syllabus and learning aims
Module aims
Compute with discrete and continuous models while keeping the underlying space, σ-algebra language at syllabus grain, and independence hypotheses explicit.
Course outline
-
01 Probability spaces 20h
Learning result Check Kolmogorov axioms on a finite space and compute a derived event.
Formative check Four-option check
-
02 Conditional probability and independence 20h
Learning result Apply Bayes or the law of total probability; give an example that is uncorrelated but not independent.
Formative check Counterexample
-
03 Discrete random variables 20h
Learning result Derive a pmf, expectation, and variance for a named discrete family.
Formative check Written derivation
-
04 Continuous random variables 20h
Learning result Use a density and a cdf, and say when a density is not a probability.
Formative check Four-option check
-
05 Expectation and variance 20h
Learning result Prove linearity of expectation without independence and derive the covariance identity for Var(X+Y); independence is sufficient, not necessary, for variance additivity.
Formative check Written derivation
-
06 Joint distributions 15h
Learning result Factor a joint pmf/pdf under independence and compute a marginal.
Formative check Written derivation
-
07 Limit theorems 20h
Learning result State LLN and CLT with hypotheses; refuse a finite-n certainty claim.
Formative check Four-option check
-
08 Synthesis 15h
Learning result Model a short experiment end to end: space, variable, expectation, approximation caveat.
Formative check Written derivation
Learning results
Specify a probability space for a named experiment, compute a distribution or expectation, and apply a limit theorem with its hypotheses.
Subject skills
Axioms, conditional probability, discrete and continuous laws, expectation, joint laws, LLN/CLT at undergraduate grain.
Transferable skills
Name the model before quoting a probability; distinguish independence from pairwise uncorrelatedness.