STATS210: pass the exams, not just read the notes
Your complete guide to University of Auckland's statistical theory course. See where the marks are, work real practice questions, and study with an AI tutor that knows STATS210.
Sia generates STATS210 practice questions, walks through probability and transformations of random variables step by step, and quizzes you on the material the exam weights most heavily.
Worked example
For a random sample from an exponential distribution with rate lambda, the maximum likelihood estimator is 1 divided by the sample mean. Is it unbiased?
State the property being tested. Unbiasedness means the estimator's expected value equals the parameter, so we need E[1/X-bar] to equal lambda.
Apply the correct reasoning. In general E[g(X)] does not equal g(E[X]) for non-linear g. For the reciprocal, Jensen's inequality gives E[1/X-bar] greater than 1/E[X-bar], so the estimator is biased upward.
Place the result in context. This is exactly why the course separates maximum likelihood estimation from the properties of estimators: MLEs are consistent and invariant, with excellent large-sample behaviour, but they are not unbiased in general.
The trap: Option C is the designed trap and the most common answer: it reasons correctly about the sample mean and then invalidly inverts. Option B asserts a property maximum likelihood estimators simply do not have. Knowing which properties MLEs do and do not possess is a core distinction in this course. classic slip!
One exam decides 60% of your grade. Dual pass: at least 50% required in this component alone. This whole page is built around that.
Overview
What STATS210 is, and where it sits
STATS 210 is the theory course sitting under everything students have previously been taught to apply. Where earlier courses give procedures — construct this interval, run that test — STATS 210 asks where those procedures come from, why they work, and what properties make one estimator better than another.
The published arc covers probability and distribution theory, likelihood as the central concept, point estimation and the properties that distinguish good estimators, interval estimation derived rather than recalled, and the theory of hypothesis testing including the construction of tests rather than merely their use.
Assessment is 60% final examination, 22% assignments, 10% test and 8% tutorials, and the pass requirement is the strictest we cover: at least 50% in the final examination alone. Where Stage I courses set that bar at 45%, this course sets it at half marks.
Always treat your own course outline and the exam timetable as authoritative.
Difficulty & time commitment
Is STATS210 hard, and how much time does it take?
STATS210 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 course.
Is this course for you
Who tends to do well, and who tends to struggle
You will likely do well if
- You are comfortable with derivation and proof, not only computation.
- You write assignment problems out fully, since the examination asks for the same style of argument.
- You keep the distinction between an estimator and an estimate clear at all times.
- You prepare for the final from week one, because 60% and the 50% threshold both sit there.
You may struggle if
- You expect to compute your way through; this course asks you to justify.
- Your calculus and algebra are shaky, since derivations expose that immediately.
- You assume properties transfer — for example that maximum likelihood estimators are unbiased.
- You rely on coursework as a safety net, which the 50% examination threshold removes.
- Redo every derivation from the definition rather than from memory. The examination rewards reconstruction, not recall.
- Keep one page mapping each estimator property — bias, consistency, efficiency, sufficiency — to how you would demonstrate it.
- Practise stating results precisely in words as well as symbols; derivation questions are marked on the argument.
- Treat the 22% of assignments as the primary rehearsal for examination technique, not as a separate task.
Syllabus
The 12 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 and distribution theory
Course descriptionThe distributional machinery the rest of the course is built on.
T2 · Transformations of random variables
Standard mathematical statistics canonDeriving the distribution of a function of a random variable.
T3 · Moment generating functions
Standard mathematical statistics canonIdentifying distributions and deriving moments and sums.
T4 · Sampling distributions derived
Course descriptionWhere the t, chi-square and F distributions actually come from.
T5 · Likelihood
Course descriptionThe central concept: reading the data as evidence about a parameter.
T6 · Maximum likelihood estimation
Course descriptionDeriving estimators from the likelihood rather than being handed them.
T7 · Properties of estimators
Course descriptionBias, consistency, efficiency, and what makes one estimator preferable.
T8 · Sufficiency
Standard mathematical statistics canonWhen a statistic carries all the information the sample holds about a parameter.
T9 · Interval estimation from theory
Course descriptionConstructing confidence intervals rather than recalling formulas.
T10 · Hypothesis testing theory
Course descriptionErrors, power, and what a test is actually doing.
T11 · Constructing tests
Course descriptionLikelihood ratio tests and the principled construction of a test.
T12 · Applying the theory
Course descriptionBringing distribution theory, likelihood and inference together on a problem.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Final examination | 60% | Final examination covering the whole course. Examination period. Dual pass: at least 50% required in this component alone. |
| Assignments | 22% | Assignments across the semester. Across the semester. Continual assessment. |
| Test | 10% | Mid-semester test. Mid-semester. Summative. |
| Tutorials | 8% | Graded tutorial work. Weekly. Continual assessment. |
- You must obtain at least 50% in the final examination alone, in addition to the overall pass requirement. This is the strictest examination threshold of any course in this network — Stage I statistics courses set it at 45%.
- Sixty percent of the grade and the 50% threshold both sit in the final examination, so this is the course with the least room for recovery that we cover. Assignments at 22% are the main protective component and the main rehearsal for the examination's derivation-style questions.
This is an exam-cram course. With the exams at 70% of the grade and the final examination alone at 60%, your result is overwhelmingly decided by how well you perform under time pressure. Dual pass: at least 50% required in this component alone.
Final exam timing: During the examination period. Confirm the exact date and venue on your exam timetable.
How to actually pass it
A weekly rhythm, two checklists, and the traps to avoid
The course 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
- Work with probability and distribution theory, including transformations of random variables.
- Use moment generating functions to identify distributions and derive moments.
- Derive the sampling distributions used in inference.
- Define likelihood and derive maximum likelihood estimators.
Before the final heaviest topics
- Assess estimators for bias, consistency, efficiency and sufficiency.
- Construct confidence intervals from theory rather than from remembered formulas.
- Explain the theory of hypothesis testing including errors and power.
- Construct a test, including likelihood ratio tests, and justify its properties.
The mistakes that cost marks
Assuming MLEs are unbiased. Maximum likelihood estimators are consistent and invariant, but not unbiased in general. This is directly examinable.
Inverting an expectation. E[1/X] is not 1/E[X]. Expectation is linear; most functions of interest are not.
Estimator confused with estimate. An estimator is a random variable with a distribution; an estimate is one realised number. Every property statement is about the former.
Answers without arguments. A correct numerical result with no derivation cannot score well in a theory course. The argument is the answer.
Formula & concept sheet
The vocabulary and formulas you must own
- Likelihood
- The probability of the observed data viewed as a function of the parameter; the central object of the course.
- Maximum likelihood estimator
- The parameter value maximising the likelihood; consistent and invariant, but not generally unbiased.
- Bias
- The difference between an estimator's expected value and the true parameter.
- Consistency
- Convergence of an estimator to the true parameter as the sample size grows.
- Efficiency
- The relative precision of an estimator, compared through variance.
- Sufficiency
- The property of a statistic that carries all the sample's information about a parameter.
- Moment generating function
- A transform identifying a distribution and yielding its moments by differentiation.
- Sampling distribution
- The distribution of a statistic, derived here rather than assumed.
- Type I and Type II error
- Rejecting a true null hypothesis, versus failing to reject a false one.
- Power
- The probability that a test correctly rejects a false null hypothesis.
- Likelihood ratio test
- A principled construction comparing likelihoods under the null and alternative hypotheses.
- Jensen's inequality
- The result relating the expectation of a convex function to the function of the expectation; why non-linear transforms introduce bias.
Common acronyms: MGF · MLE · MSE · UMVUE.
Where it fits
Prerequisites, related courses & why it matters
Prerequisite published by the department: STATS 125 and 15 points from MATHS 208, 250 or equivalent. STATS 210 is worth 15 points and is offered in both semesters. It is the gateway to the Stage III theoretical statistics courses and to postgraduate study in statistics.
Your STATS210 study toolkit
Study the course with Sia, not just read about it
Each tool already knows STATS210: your syllabus, your texts, and where the marks are. Grouped by how you study, from first contact to exam week.
FAQ
Frequently asked questions
Is STATS 210 hard?
It rates hard — the highest in our New Zealand set. It is mathematically demanding, 60% of the grade sits in the final examination, and the requirement of at least 50% in that examination alone is the strictest we cover.
What is the assessment breakdown?
Final examination 60%, assignments 22%, test 10%, tutorials 8%. You must obtain at least 50% in the final examination alone.
Why is the threshold 50% rather than 45%?
The department sets it higher for this course than for Stage I statistics. In practice it means the examination must be passed on its own terms; no configuration of coursework marks compensates.
What do I need before taking it?
STATS 125 and 15 points from MATHS 208, 250 or equivalent. The mathematics prerequisite is real — this course derives results rather than applying them.
How is it different from earlier statistics courses?
Earlier courses teach you to use procedures. This one derives them. You will prove why an estimator has a property rather than being told that it does.
What is the hardest part?
For most students it is the shift from computation to derivation, concentrated in the properties of estimators and the construction of tests. Practising written derivations, not just numerical answers, is what closes that gap.
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