MAST90105: pass the exams, not just read the notes
Your complete guide to University of Melbourne's methods of mathematical statistics unit. See where the marks are, work real practice questions, and study with an AI tutor that knows MAST90105.
Sia generates MAST90105 practice questions, works through them step by step, and quizzes you on the material the exam weights most heavily.
Worked example
You observe n independent Bernoulli(p) trials with k successes. What is the maximum likelihood estimator of p?
The likelihood is L(p) = p^k (1 − p)^(n − k).
Differentiate and set to zero: k/p − (n − k)/(1 − p) = 0.
Solving gives p_hat = k / n, the sample proportion of successes.
The trap: Guessing p_hat = k (the raw count) or inverting to n/k. The maximum likelihood estimator maximises the log-likelihood, which gives the sample proportion k/n; the (k+1)/(n+2) form is a Bayesian posterior mean, not the MLE. classic slip!
One exam decides 35% of your grade. This whole page is built around that.
Overview
What MAST90105 is, and where it sits
MAST90105 Methods of Mathematical Statistics is the University of Melbourne's postgraduate mathematical-statistics subject, taught in the School of Mathematics and Statistics. It builds statistical inference from its probability foundations: probability and Bayes' theorem, discrete and continuous random variables and moment-generating functions, the central limit theorem, bivariate distributions and correlation, transformations and sampling distributions, point estimation by the method of moments and maximum likelihood, estimator properties and the Cramér-Rao lower bound, interval estimation via pivotal quantities, hypothesis testing and the Neyman-Pearson lemma, and distribution-free and categorical methods.
The subject is theoretically rigorous and quantitative, with an applied component in R. Most of the grade is exam-based, rewarding the ability to derive estimators and tests and prove their properties rather than apply canned procedures. The recurring skill is moving fluently between probability theory, estimation and the formal logic of hypothesis testing.
Official outline: handbook.unimelb.edu.au · MAST90105 outline. Always treat the official outline and the exam timetable as authoritative.
Difficulty & time commitment
Is MAST90105 hard, and how much time does it take?
MAST90105 is manageable if you keep a weekly rhythm and treat the back half as the main event. The pattern is consistent: it starts gently and steepens, and the heaviest assessment is the part that separates grades.
The difficulty curve and the assessment weighting point the same way: the back half is harder and worth more. Front-loading effort there is the highest-return decision in the unit.
Is this unit for you
Who tends to do well, and who tends to struggle
You will likely do well if
- You are comfortable with probability theory, calculus and formal proof, since the subject derives rather than applies.
- You practise deriving estimators and tests by hand, not just running them in R.
- You keep pace through the estimation-and-inference block, the steepest part.
You may struggle if
- You are shaky on probability and calculus, which the estimation theory assumes.
- You treat statistics as recipes; this subject asks you to prove why methods work.
- You under-practise the maximum-likelihood, Cramér-Rao and Neyman-Pearson material.
- Derive the standard MLEs (Bernoulli, normal, Poisson) from the log-likelihood until it is routine.
- Practise the Cramér-Rao bound and Neyman-Pearson lemma as proofs, not formulas.
- Work past exams by hand and use R only to check, since the exams reward derivation.
Syllabus
The 10 topics, topic by topic
The exam-weight marker on each topic shows where the marks concentrate. The amber topics carry the highest exam weight.
T1 · Probability Foundations and Bayes
Axioms & set identities · counting · conditional probability · total probability & <b>Bayes’ theorem</b> · independence
T2 · Discrete Random Variables and MGFs
PMF / CDF · expectation & variance · <b>moment generating functions</b> · the discrete families
T3 · Continuous Random Variables and the CLT
Density / CDF · mean, median, variance · exponential / gamma / χ² · <b>normal</b> · CLT & LLN
T4 · Bivariate Distributions and Correlation
Joint / marginal / conditional · covariance & <b>correlation</b> · the <b>bivariate normal</b> · zero-corr ≠ independence
T5 · Transformations and Sampling Distributions
<b>CDF method</b> · Jacobian / change of variables · <b>MGF method</b> · <b>Z → t → χ² → F</b>
T6 · Point Estimation: Method of Moments and Maximum Likelihood
<b>Method of moments</b> · the likelihood · <b>maximum likelihood</b> — the full L → ln L → score = 0 recipe
T7 · Estimator Properties and the Cramér–Rao Lower Bound
Bias, variance & <b>MSE</b> · <b>Fisher information</b> · the <b>Cramér–Rao lower bound</b> · regularity conditions · asymptotic normality
T8 · Interval Estimation and Pivotal Quantities
<b>Pivotal quantities</b> → CIs for means & proportions · large-sample CIs · distribution-free CIs for percentiles
T9 · Hypothesis Testing and Neyman–Pearson
Type I / II & power · the <b>Neyman–Pearson</b> lemma & LRTs · one-sample <b>t</b>, variance <b>χ²</b>, proportion <b>z</b>
T10 · Distribution-Free and Categorical Methods
Sign & rank tests · <b>goodness-of-fit χ²</b> · <b>contingency tables</b> & tests of independence
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Mid-semester exam | 35% | Mid-semester · on-campus, invigilated, <b>3 hours</b>; the <b>probability half</b> (Weeks 1–7) · <b>one A4 double-sided sheet</b> + Casio FX-82 · <b>distribution table provided</b>. |
| Final exam | 35% | End of semester · on-campus, invigilated, <b>3 hours</b>; cumulative, weighted to the <b>inference half</b> (Weeks 8–12) · <b>one A4 sheet</b> + Casio FX-82 · <b>distribution table provided</b>. |
| Written assignments | 20% | Four individual written assignments across the semester (~5% each). |
| Computer / R lab test | 10% | In a lab on a laptop, <b>open-book</b>; R-based; no communication — the only open-book task. |
- Pass on a weighted average of at least 50%. No single-component hurdle unless noted; confirm against the official subject page.
This is an exam-cram unit. With the exams at 80% of the grade and the mid-semester exam alone at 35%, your result is overwhelmingly decided by how well you perform under time pressure.
How to actually pass it
A weekly rhythm, two checklists, and the traps to avoid
The unit rewards consistency over cramming, and practice over re-reading. Here is the loop that works, then what to have nailed before each exam.
The weekly loop
Before the mid-semester checklist
Before the final heaviest topics
- Rehearse maximum likelihood and method-of-moments estimation for standard distributions.
- Drill the Cramér-Rao lower bound and estimator-property proofs.
- Practise Neyman-Pearson hypothesis tests and interval estimation via pivotal quantities.
- Work both exams by hand under time pressure across probability and inference.
The mistakes that cost marks
Applying recipes without deriving. The subject rewards derivation and proof. Running R procedures without understanding the theory leaves you unable to answer the derivation questions that carry the marks.
Confusing MLE and Bayesian estimates. The MLE maximises the likelihood (k/n for Bernoulli); posterior means like (k+1)/(n+2) are different estimators. Mixing them is a common inference error.
Cramming the inference block. MLE, Cramér-Rao and Neyman-Pearson are the steepest, most heavily examined topics and do not compress into the final week.
Teaching team
Who teaches MAST90105
No teaching staff are publicly listed for this offering. Check the official course page for the current coordinator and lecturers.
Formula & concept sheet
The vocabulary and formulas you must own
- Likelihood and MLE
- The likelihood L(theta) is the joint density viewed as a function of the parameter; the maximum likelihood estimator maximises L (or its log). For Bernoulli(p) it gives k/n.
- Method of moments
- Estimates parameters by equating sample moments to population moments and solving; a simpler alternative to maximum likelihood.
- Cramér-Rao lower bound
- A lower bound on the variance of any unbiased estimator, equal to the inverse of the Fisher information; an estimator achieving it is efficient.
- Neyman-Pearson lemma
- For testing two simple hypotheses, the likelihood-ratio test is the most powerful test at a given significance level.
- Central limit theorem
- The sample mean of n independent identically distributed variables is approximately normal for large n, regardless of the underlying distribution.
Common acronyms: MLE · MoM · CLT · MGF · CRLB · NP · iid.
Where it fits
Prerequisites, related units & why it matters
Postgraduate mathematical-statistics subject; assumes probability, calculus and prior statistics. Check the UniMelb Handbook for prerequisites.
Your MAST90105 study toolkit
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FAQ
Frequently asked questions
Is MAST90105 hard?
It is moderate-to-hard: a rigorous postgraduate mathematical-statistics subject that derives estimators and tests and proves their properties. Strong probability and calculus and consistent hand derivation make it manageable, but it is theory-heavy and exam-weighted.
How is MAST90105 assessed?
A 35% mid-semester exam, a 35% final exam, written assignments worth 20%, and a computer/R lab test worth 10%. The components sum to 100% and the two exams dominate.
What does it cover?
Probability and Bayes, random variables and moment-generating functions, the central limit theorem, estimation by method of moments and maximum likelihood, the Cramér-Rao bound, interval estimation, Neyman-Pearson hypothesis testing, and distribution-free methods.
How much maths is involved?
A lot: it is mathematical statistics, requiring probability theory, calculus and formal derivation and proof, with an applied component in R.
What background do I need?
Solid probability, calculus and ideally prior statistics. The subject builds inference rigorously from probability foundations.
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