Probability & Statistical Inference

This module equips learners with the probability theory and inferential machinery needed to design, analyse and critique quantitative studies. Starting from axiomatic probability and the common families of distributions, learners progress through estimation, confidence intervals, hypothesis testing and the Bayesian paradigm. Mastery is demonstrated when a learner can pick an appropriate inferential procedure for a given design, state its assumptions, carry it out correctly and interpret the result without overclaiming significance or certainty.

Learning Outcomes

Apply the probability axioms, conditional probability and Bayes' theorem to structured problems.

Identify and work with the binomial, Poisson, geometric, normal, exponential, gamma, beta, chi-square, t and F distributions.

Derive and interpret expectations, variances, covariances and moment generating functions.

Explain and apply the law of large numbers and the central limit theorem.

Construct point estimators (MLE, MoM) and evaluate bias, variance, consistency and efficiency.

Build confidence intervals and conduct z, t, chi-square and F hypothesis tests under their assumptions.

Compute p-values, power and effect size and interpret them without binary significance thinking.

Compare frequentist and Bayesian inference, including conjugate updating and credible intervals.

Topics Covered

Probability axioms, sample spaces and conditional probability

Bayes' theorem and the law of total probability

Discrete distributions: Bernoulli, binomial, geometric, Poisson, hypergeometric

Continuous distributions: uniform, exponential, normal, gamma, beta

Expectation, variance, covariance and correlation

Moment generating and characteristic functions

Joint, marginal and conditional distributions; independence

Transformations of random variables and the Jacobian

Convergence in probability and in distribution; LLN and CLT

Point estimation: MoM, MLE, bias, MSE, consistency, Fisher information

Confidence intervals: z, t, exact binomial, Wilson

Hypothesis testing: one- and two-sample z and t tests, paired tests

Chi-square tests of goodness of fit and independence; F tests for variance and ANOVA

Non-parametric and resampling methods (sign, Wilcoxon, bootstrap, permutation)

Bayesian inference: priors, posteriors, conjugacy and credible intervals

Skills Developed

Probability modelling Distributional reasoning Inference procedure selection Interval construction p-value and effect size interpretation Likelihood and MLE derivation Bayesian updating Statistical software fluency

Suggested Resources

Casella, G. & Berger, R.L. (2002). Statistical Inference, 2nd ed., Duxbury

Wasserman, L. (2004). All of Statistics, Springer

DeGroot, M.H. & Schervish, M.J. (2012). Probability and Statistics, 4th ed., Pearson

Hogg, R.V., McKean, J.W. & Craig, A.T. (2019). Introduction to Mathematical Statistics, 8th ed., Pearson

Gelman, A. et al. (2013). Bayesian Data Analysis, 3rd ed., Chapman & Hall

Grinstead, C.M. & Snell, J.L. (1997). Introduction to Probability, AMS

MIT OpenCourseWare 18.650 Statistics for Applications

Assessment

Learners are judged on choosing and executing the right inferential procedure and interpreting it honestly. The weighting rewards practice with real data over mechanical calculation.

Written problem set portfolio across six sub-topics35%

Analysis report on a provided dataset30%

Timed closed-book examination20%

MCQ diagnostic bank (this module's 200+ item pool)15%

Practice bank: 195 MCQs · Timed mock & self-paced modes

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