STATS225: pass the exams, not just read the notes
Your complete guide to University of Auckland's mathematical statistics course. See where the marks are, work real practice questions, and study with an AI tutor that knows STATS225.
Sia generates STATS225 practice questions, walks through discrete random variables and continuous random variables step by step, and quizzes you on the material the exam weights most heavily.
Worked example
X and Y are independent standard normal variables. What is the distribution of X + Y, and why can you state it without integrating?
Use the closure property. The normal family is closed under addition of independent members: a sum of independent normal variables is itself normal. That fact alone fixes the shape without any integration.
Get the variance from independence. Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y), and independence makes the covariance zero, so the variance is 1 + 1 = 2.
Note what the classical restriction buys you. The course concentrates on the normally distributed case precisely because results like this are available in closed form. Option D is what you would face outside the normal family — and it is why the restriction exists.
The trap: Option C confuses the sum with the sum of squares. X squared plus Y squared is chi-square with two degrees of freedom; X plus Y is normal. Both results appear in this course and they are constantly swapped. Option B forgets that variances add while standard deviations do not. classic slip!
One exam decides 75% of your grade. Summative. This whole page is built around that.
Overview
What STATS225 is, and where it sits
STATS 225 is described by the department as an introduction to the mathematical theory of statistics that prepares students for further study in the subject, and it states plainly that the course concentrates on the classical, normally distributed case. If you are considering postgraduate work in statistics, the department says this course will be very useful to you.
The published topics are specific: discrete and continuous random variables, expected values, multivariate distribution theory, multivariate transformations, sampling theory, estimation and testing. The centre of gravity is the multivariate material — moving from one random variable to several, and deriving what happens when you transform them.
The assessment structure is the most examination-weighted in the entire network: assignments 20%, in-class test 5%, final examination 75%. There is no separate examination threshold published for this course, unlike STATS 101, 125 and 210, but with three quarters of the grade in one paper the practical effect is similar.
Always treat your own course outline and the exam timetable as authoritative.
Difficulty & time commitment
Is STATS225 hard, and how much time does it take?
STATS225 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 multivariable calculus, since multivariate transformations require Jacobians.
- You treat assignments as examination rehearsal; at 20% they are the only real buffer against a 75% paper.
- You know the closure and distributional results well enough to avoid deriving them under time pressure.
- You keep the sum and the sum of squares of normals clearly apart.
You may struggle if
- Your calculus is shaky; the transformation material exposes it immediately.
- You rely on the in-class test to gauge your standing — at 5% it barely moves your grade.
- You memorise distributional results without knowing when each applies.
- You leave examination preparation late, with three quarters of the grade in one sitting.
- Build a results sheet of closure properties: which families are closed under addition, scaling and transformation.
- Practise the Jacobian method until the mechanics are automatic; the difficulty is bookkeeping, not concept.
- For every named distribution, know what it is the distribution of — that is what tells you when to reach for it.
- Work full past papers under time. At 75%, examination technique is the single highest-return preparation available.
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 · Discrete random variables
Course descriptionMass functions and the standard discrete families in a theoretical treatment.
T2 · Continuous random variables
Course descriptionDensity functions, cumulative distribution functions and continuous families.
T3 · Expected values
Course descriptionExpectation as an operator, and its algebra.
T4 · Variance, covariance and moments
Standard mathematical statistics canonSecond-order structure and higher moments.
T5 · Joint and marginal distributions
Course description (multivariate)Distributions of several random variables together, and recovering the parts.
T6 · Conditional distributions and independence
Course description (multivariate)Conditioning in the multivariate setting, and what independence means there.
T7 · The multivariate normal distribution
Course description (classical case)The distribution the course is built around, and why it is tractable.
T8 · Multivariate transformations
Course descriptionDeriving the distribution of a function of several random variables, including the Jacobian method.
T9 · Sampling theory
Course descriptionThe distribution of statistics computed from a sample.
T10 · Estimation
Course descriptionPoint estimation and the properties of estimators in the classical case.
T11 · Hypothesis testing
Course descriptionTesting in the normally distributed setting.
T12 · Preparing for postgraduate statistics
Departmental statement of purposeHow the theory here underpins later study.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Final examination | 75% | Final examination covering the whole course. Examination period. Summative. |
| Assignments | 20% | Assignments across the semester. Across the semester. Continual assessment. |
| In-class test | 5% | In-class test. Mid-semester. Summative. |
- The three components sum to 100. The department publishes no separate examination threshold for this course, unlike several of its other statistics offerings.
- At 75%, this is the most examination-dominated course in the network. The 5% in-class test is diagnostic rather than protective, and 20% of assignments is the only meaningful buffer — which makes the assignments the primary rehearsal for the examination rather than a side task.
This is an exam-cram course. With the exams at 80% of the grade and the final examination alone at 75%, your result is overwhelmingly decided by how well you perform under time pressure. Summative.
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 discrete and continuous random variables in a theoretical treatment.
- Compute expected values and apply the algebra of expectation.
- Handle joint, marginal and conditional distributions.
- Explain the multivariate normal distribution and its properties.
Before the final heaviest topics
- Apply multivariate transformations, including the Jacobian method.
- Derive sampling distributions in the classical case.
- Carry out estimation and justify estimator properties.
- Construct and apply hypothesis tests in the normally distributed setting.
The mistakes that cost marks
Sum confused with sum of squares. X + Y of independent standard normals is normal; X squared plus Y squared is chi-square. Both appear in this course.
Adding standard deviations. Variances add for independent variables; standard deviations do not.
Jacobian omitted or inverted. A multivariate transformation without the correct Jacobian factor gives a function that is not a density.
Treating the 5% test as a safety net. It is diagnostic. The assignments at 20% are the only meaningful buffer against a 75% examination.
Formula & concept sheet
The vocabulary and formulas you must own
- Random variable
- A function assigning a number to each outcome, described by a mass or density function.
- Expected value
- The probability-weighted average of a random variable; a linear operator.
- Joint distribution
- The distribution of several random variables considered together.
- Marginal distribution
- The distribution of one variable obtained by summing or integrating out the others.
- Conditional distribution
- The distribution of one variable given a value of another.
- Multivariate normal
- The multivariate generalisation of the normal distribution; the classical case this course concentrates on.
- Covariance matrix
- The array of pairwise covariances that, with the mean vector, specifies a multivariate normal.
- Jacobian
- The determinant factor correcting for the change of volume in a multivariate transformation.
- Closure property
- The fact that a family of distributions is preserved under an operation, such as normals under addition.
- Sampling distribution
- The distribution of a statistic computed from a sample.
- Estimator
- A rule for producing a parameter estimate from data; a random variable with its own distribution.
- Chi-square distribution
- The distribution of a sum of squared independent standard normal variables.
Common acronyms: CDF · MGF · MVN · PDF.
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. The course is worth 15 points. The department states it is designed to prepare students for further study in statistics and is particularly useful for those considering postgraduate work.
Your STATS225 study toolkit
Study the course with Sia, not just read about it
Each tool already knows STATS225: 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 225 hard?
It rates hard. The mathematics is demanding, particularly the multivariate transformations, and 75% of the grade sits in the final examination — the highest single weighting in this network.
What is the assessment breakdown?
Final examination 75%, assignments 20%, in-class test 5%.
Is there an examination threshold?
The department publishes none for this course, unlike STATS 101, 125 and 210. With 75% of the grade in the examination the practical effect is similar, but there is no separate hurdle to clear.
What do I need before taking it?
STATS 125 and 15 points from MATHS 208, 250 or equivalent.
How does it differ from STATS 210?
Both are theory courses. STATS 225 concentrates on the classical normally distributed case and emphasises multivariate distribution theory and transformations; STATS 210 centres on likelihood, estimator properties and the construction of tests.
Should I take it for postgraduate study?
The department says so directly: if you are considering postgraduate work in statistics, this course will be very useful to you.
Study STATS225 with Sia
Work through discrete random variables, continuous random variables, expected values and the rest of the course with a tutor that knows it and quizzes you on the topics the assessments weight most heavily.
Start studying with Sia
