MAST90105 · Methods Of Mathematical Statistics
The Complete Exam Bible · S1 2026Methods of Mathematical Statistics
Methods of Mathematical Statistics is the University of Melbourne's graduate gateway to statistical theory — it climbs the ladder from probability (axioms and Bayes, discrete and continuous random variables, moment generating functions, the named distribution families, the bivariate normal, and how transforming or sampling from a normal manufactures the t, χ² and F distributions) into inference (the method of moments and maximum likelihood, the bias–variance–MSE grading of an estimator and the Cramér–Rao lower bound, pivotal-quantity confidence intervals, the Neyman–Pearson lemma and likelihood-ratio tests, and the χ² methods for counts). 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.
What MAST90105 covers
Ten teaching chapters across two halves — probability then inference — one exam-ready map. Each links to its free chapter guide.
How MAST90105 is assessed
| Component | Weight | Format |
|---|---|---|
| Mid-semester exam | 35% | Mid-semester · on-campus, invigilated, 3 hours; the probability half (Weeks 1–7) · one A4 double-sided sheet + Casio FX-82 · distribution table provided |
| Final exam | 35% | End of semester · on-campus, invigilated, 3 hours; cumulative, weighted to the inference half (Weeks 8–12) · one A4 sheet + Casio FX-82 · distribution table provided |
| Written assignments | 20% | Four individual written assignments across the semester (~5% each) |
| Computer / R lab test | 10% | In a lab on a laptop, open-book; R-based; no communication — the only open-book task |
The maximum-likelihood recipe — L → ln L → score = 0, mark by mark
- +1Name the method. We want the value of λ that makes the observed sample most likely — maximum likelihood. Write the likelihood, take logs, set the score to zero, then check the second derivative.
- +1Likelihood: for the i.i.d. sample, L(λ) = ∏ λe−λxi = λn e−λ∑xi.
- +1Log-likelihood: ℓ(λ) = ln L = n ln λ − λ∑xi. Logs turn the product into a sum that is easy to differentiate.
- +1Score = 0: ℓ′(λ) = n/λ − ∑xi = 0 ⇒ λ̂ = n / ∑xi = 1/x̄.
- +1Confirm a maximum: ℓ″(λ) = −n/λ2 < 0 for all λ > 0, so the stationary point is a maximum.
- +1Interpret: the MLE of the rate is the reciprocal of the sample mean — intuitive, since 1/λ is the exponential's mean. (−ℓ″ evaluated at λ̂ is the observed information, the input to the standard error.)
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)
- MX(t) = E[etX]; 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
Is MAST90105 hard?
It is a rigorous, derivation-heavy graduate subject: most marks reward setting up the right method — an MLE, a Fisher-information bound, a pivot, a test statistic — and carrying the algebra correctly under exam time. The difficulty is mastering the recipes so they are automatic, because the same handful (transformations, the MLE, the CRLB, pivots, hypothesis tests) recur on fresh parameters. A solid grounding in calculus and probability makes it very manageable.
How is MAST90105 assessed?
By two on-campus 3-hour written exams — a mid-semester on the probability half (Weeks 1–7, 35%) and a cumulative final weighted to the inference half (Weeks 8–12, 35%) — plus four written assignments (~20% combined) and a 10% open-book R lab test. Into each written exam you may carry one A4 double-sided sheet and a non-programmable Casio FX-82, and a distribution table is provided on the paper. Confirm this year's exact weights, dates and conditions on your own LMS.
Is the exam open or closed book, and what does the distribution table give me?
The two written exams are neither fully open nor fully closed book: you may bring one A4 double-sided handwritten or printed sheet plus a Casio FX-82, and a distribution table listing every named distribution's PMF/PDF, MGF, mean and variance is appended to the paper. So your A4 sheet should never copy that table — spend it on the derivation recipes and decision logic the table cannot give you. (The R lab test is the only fully open-book task.)
What should I put on my A4 sheet?
The things the provided table omits: the MLE recipe (L → ln L → score = 0), the Cramér–Rao regularity conditions and when they fail, the Z → t → χ² → F map, the conjugate-prior table, and your library of pivotal quantities. The PMF, MGF, mean and variance for every distribution are already on the paper — copying them wastes the sheet. Recipes win marks; the table just supplies the constants.
Is using AskSia for MAST90105 cheating?
No. AskSia is a study reference written in our own words — we host none of your lecturer's slides, assignments or past papers, and every worked example uses our own invented stems, parameters and numbers, never the assessed questions. Sia teaches you the method to earn the marks; it does not complete or sit your assessments.
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.
