MAST90105 Methods of Mathematical Statistics

How MAST90105 is assessed

What MAST90105 covers

Ten teaching chapters across two halves — probability then inference — one exam-ready map. Each links to its free chapter guide.

The subject is assessed by two 3-hour written exams — a mid-semester on the probability half and a cumulative final weighted to inference — into each of which you carry one A4 double-sided sheet and a non-programmable Casio FX-82, with a distribution table printed on the paper.

That single fact reshapes how you study: the table hands you every PMF, PDF, MGF, mean and variance, so the marks live in which method to reach for and how to set it up. This guide drills exactly that — for every prompt: name the method → set up the recipe → compute → read the constants off the provided table.

Worked example · free

The maximum-likelihood recipe — L → ln L → score = 0, mark by mark

Key terms

Maximum-likelihood estimator (MLE)

The parameter value that maximises the likelihood of the observed sample. For a regular model you find it by the recipe L → ln L → score = 0, then confirm a maximum with the second derivative. MLEs are consistent and asymptotically normal, and under regularity conditions achieve the Cramér–Rao lower bound in large samples.

Fisher information

A measure of how sharply the log-likelihood peaks around the true parameter: I(θ) = −E[ℓ″(θ)]. More information means a more identifiable parameter and a smaller achievable variance — its reciprocal, 1/[nI(θ)], is the Cramér–Rao lower bound for an unbiased estimator.

Pivotal quantity

A function of the data and the parameter whose distribution does not depend on the parameter — for a normal mean, (X̄ − μ)/(S/√n) is t with n−1 degrees of freedom. Because its distribution is known, you can bracket it between quantiles and invert the inequality to produce a confidence interval. One recipe generates every standard CI on the course.

Neyman–Pearson lemma

For testing a simple null against a simple alternative, the most powerful test at a given significance level rejects when the likelihood ratio L(θ₁)/L(θ₀) exceeds a threshold. It is the optimality result behind likelihood-ratio tests and the reason the usual t, χ² and z statistics are the right ones.

Moment generating function (MGF)

M_X(t) = E[e^tX]; its derivatives at t = 0 deliver the moments, and because the MGF determines the distribution uniquely, matching MGFs is a clean way to name the distribution of a sum or a transformation. The MGF method is one of the three recipes for finding the law of Y = g(X).

MAST90105 FAQ

Questions students ask in each MAST90105 chapter

What is the difference between mutually exclusive and independent?

They are opposites, not synonyms. Mutually exclusive means the events cannot both happen, P(A∩B) = 0, so if one occurs the other cannot — that is maximal dependence. Independent means P(A∩B) = P(A)P(B), so one happening tells you nothing about the other. Two events with positive probability cannot be both mutually exclusive and independent. Chapter 01 · Probability Foundations and Bayes

Why bother with the MGF when the table gives me the mean and variance?

Two reasons the table cannot replace. First, the MGF identifies the distribution of a sum of independent variables by multiplying MGFs — the fastest way to show a sum of Poissons is Poisson, or a sum of normals is normal. Second, it is a uniqueness tool: if two variables share an MGF they share a distribution. The table gives formulas; the MGF gives derivations and the recipe for sums. Chapter 02 · Discrete Random Variables and MGFs

Why is P(X = x) = 0 for a continuous variable?

Because probability is the area under the density, and a single point has zero width and therefore zero area. Only intervals carry probability. A practical consequence: for continuous variables P(X ≤ a) and P(X < a) are equal, so you never have to fuss over a strict-versus-weak inequality at an endpoint — unlike the discrete case. Chapter 03 · Continuous Random Variables and the CLT

Does zero correlation mean two variables are independent?

No — this is the most over-generalised result in the topic. Correlation captures only linear association, so two variables related through a curve (for example Y = X² with X symmetric about zero) can have ρ = 0 while being completely dependent. Independence is the stronger condition that the joint law factors into the marginals. The one exception is the bivariate normal, where zero correlation does force independence. Chapter 04 · Bivariate Distributions and Correlation

Which transformation method should I use?

Use the CDF method as your default — it works for any g, including non-monotone maps, where you split the event into pieces. Use the Jacobian method when g is one-to-one and monotone, for speed. Use the MGF method when you need to name the result, especially for a sum of independent variables, since matching MGFs identifies the family with no integration. Knowing which to reach for is itself examined. Chapter 05 · Transformations and Sampling Distributions

When do the method of moments and maximum likelihood disagree?

For many simple one-parameter models (Bernoulli, Poisson, exponential) they coincide, because matching the mean is the same as solving the score. They diverge for models where the moment equations and the likelihood pull in different directions — for example skewed or multi-parameter families, or boundary models like Uniform(0,θ). When they differ, the MLE is generally preferred for its better large-sample efficiency, with MoM useful as a quick starting value. Chapter 06 · Point Estimation: Method of Moments and Maximum Likelihood

Why prefer an estimator with lower MSE over an unbiased one?

Because MSE = variance + bias² is what actually measures closeness to the truth. An unbiased estimator can have a large variance, and a slightly biased estimator with much smaller variance can have a lower MSE and so be closer on average. Unbiasedness is a desirable property, not the goal; minimising MSE (or finding the minimum-variance unbiased estimator) is the real objective. Chapter 07 · Estimator Properties and the Cramér–Rao Lower Bound

What does ‘95% confidence’ actually mean?

It is a statement about the procedure, not the parameter. If you repeated the sampling and interval construction many times, about 95% of the resulting intervals would contain the true parameter. For any one observed interval the parameter is either in it or not — there is no 95% probability attached to that fixed interval. This is the most-tested interpretation point in the chapter. Chapter 08 · Interval Estimation and Pivotal Quantities

What is the difference between the significance level and the power?

The significance level α is the probability of a Type I error — rejecting a true null — which you fix in advance, commonly at 5%. Power is 1 − β, the probability of correctly rejecting a false null. You control α by the cut-off; power then depends on the true effect size, the sample size and the variability. For a fixed n there is a trade-off: shrinking α reduces power. Chapter 09 · Hypothesis Testing and Neyman–Pearson

When should I use a distribution-free test?

When the parametric assumptions behind a t or z test are doubtful — small samples with clear non-normality, heavy skew, or ordinal data where only ranks are meaningful. Sign and rank tests then give valid inferences using only the ordering of the observations. The trade-off is some loss of power when the data actually are normal, so prefer the parametric test when its assumptions hold. Chapter 10 · Distribution-Free and Categorical Methods

How to study for the exam

Study MAST90105 as two halves that share one habit.

The probability half (Weeks 1–7) builds the machinery — MGFs, the distribution families, the bivariate normal, and the transformations that manufacture the t, χ² and F laws — and the inference half (Weeks 8–12) consumes it: you cannot derive an MLE without a likelihood, bound its variance without Fisher information, build a CI without a pivot, or run a t-test without the sampling distribution of the mean.

Chapters 6–7 — the MLE and the CRLB — are where most final-exam marks are won and lost, so slow down there. For every prompt run the same loop: name the method → write the recipe before the numbers → carry the algebra → read the constants off the provided table.

Because the table hands you every formula, build your A4 sheet around the recipes and decision rules it cannot give you, and drill the recurring exam prompt-types until the method jumps out from the cue.