STATS125: pass the exams, not just read the notes
Your complete guide to University of Auckland's probability and its applications course. See where the marks are, work real practice questions, and study with an AI tutor that knows STATS125.
Sia generates STATS125 practice questions, walks through probability axioms and conditional probability step by step, and quizzes you on the material the exam weights most heavily.
Worked example
A Markov chain has states A and B, with P(A to B) = 0.4 and P(B to A) = 0.2. What is the long-run proportion of time spent in state A?
Set up the stationary distribution. Write the long-run proportions as pi_A and pi_B with pi_A + pi_B = 1. Stationarity means the flow into each state balances the flow out of it.
Solve with the normalising condition. Substituting into pi_A + pi_B = 1 gives 3 pi_A = 1, so pi_A = 1/3 and pi_B = 2/3.
Sanity check the direction. The chain leaves A twice as readily as it leaves B, so it should spend less time in A. One third is below one half, which is consistent. Option B is what you get by solving correctly and then reporting the wrong state.
The trap: Reading the transition probability as the long-run proportion — option C takes 0.4 straight from the question. A transition probability describes one step from a given state; the stationary distribution describes where the chain sits over the long run. They are different objects and the question is built to test that. classic slip!
One exam decides 50% of your grade. Dual pass: at least 45% required in this component alone. This whole page is built around that.
Overview
What STATS125 is, and where it sits
STATS 125 is the University of Auckland's mathematical probability course, and it should not be confused with the applied statistics courses at the same stage. Where STATS 101 teaches reasoning with data, this course develops probability as a mathematical structure and then applies it to random processes.
The topic arc runs from the axioms and combinatorial counting through conditional probability and independence, then random variables and their distributions, expectation and variance, joint distributions, into probability generating functions, the limit theorems, and finally Markov chains as the worked application of the whole toolkit.
Assessment is 50% final examination, 20% test, 18% assignments and 12% tutorials. Two features matter for planning: the tutorial component is graded, which is unusual and effectively converts weekly preparation into marks, and the dual pass rule requires at least 45% in the final examination alone regardless of your total.
Always treat your own course outline and the exam timetable as authoritative.
Difficulty & time commitment
Is STATS125 hard, and how much time does it take?
STATS125 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
- Your calculus is fluent; continuous distributions require integration without hesitation.
- You work problems rather than reading solutions, since every topic here is a technique.
- You attend and prepare for tutorials, which carry 12% and are the easiest marks in the course.
- You track your projected final examination mark, not just your total, because of the 45% threshold.
You may struggle if
- You expect STATS 101's approach; this course is mathematical and moves quickly.
- You memorise distributions without understanding what generating functions do for you.
- You treat Markov chains as matrix arithmetic without grasping the Markov property.
- You rely on coursework to carry a weak final, which the dual pass rule specifically prevents.
- Build one card per distribution: parameters, mass or density function, mean, variance, and the situation it models.
- Learn generating functions as a machine — put a distribution in, get moments out. Practising the mechanics makes the abstraction concrete.
- For Markov chains, always write the transition matrix first and check every row sums to one before doing anything else.
- Sit past finals under time; the 45% threshold makes examination technique a pass-or-fail matter, not a grade matter.
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 axioms and counting
Course descriptionSample spaces, the axioms, and combinatorial methods for counting outcomes.
T2 · Conditional probability and independence
Course descriptionConditioning, the multiplication rule, total probability and Bayes' theorem.
T3 · Discrete random variables
Course descriptionProbability mass functions and the standard discrete families.
T4 · Continuous random variables
Course descriptionDensity functions, cumulative distribution functions and continuous families.
T5 · Expectation and variance
Course descriptionExpected value, variance, and the algebra of expectation operators.
T6 · Joint distributions
Course descriptionTwo-dimensional random variables, marginals and conditional distributions.
T7 · Independence and covariance
Course descriptionWhen random variables are independent, and how covariance measures dependence.
T8 · Probability generating functions
Course descriptionEncoding a distribution in a function, and reading moments back out of it.
T9 · Sums of random variables
Standard probability canonDistributions of sums, and why generating functions make them tractable.
T10 · Limit theorems
Course descriptionThe law of large numbers and the central limit theorem as statements about long-run behaviour.
T11 · Markov chains: structure
Course descriptionStates, transition matrices and the Markov property.
T12 · Markov chains: long-run behaviour
Course descriptionClassification of states, stationary distributions and limiting behaviour.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Final examination | 50% | Final examination covering the whole course. Examination period. Dual pass: at least 45% required in this component alone. |
| Test | 20% | Mid-semester test. Mid-semester. Summative. |
| Assignments | 18% | Assignments across the semester. Across the semester. Continual assessment. |
| Tutorials | 12% | Graded tutorial work across the semester. Weekly. Continual assessment. |
- You must obtain at least 50% overall AND at least 45% in the final examination alone. Coursework, however strong, cannot compensate for a final below that threshold.
- The dual pass rule is the binding constraint: 45% in the final examination is required independently of your total. Note also that tutorials carry 12% — small individually, but they are the most reliably earned marks in the course and they reward keeping pace rather than catching up.
This is an exam-cram course. With the exams at 70% of the grade and the final examination alone at 50%, your result is overwhelmingly decided by how well you perform under time pressure. Dual pass: at least 45% 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
- Apply the probability axioms and combinatorial counting.
- Use conditional probability, independence and Bayes' theorem.
- Work with discrete and continuous random variables and their distributions.
- Compute expectation and variance and apply their properties.
Before the final heaviest topics
- Handle joint distributions, independence and covariance.
- Use probability generating functions to obtain moments and distributions of sums.
- State and apply the law of large numbers and the central limit theorem.
- Analyse a Markov chain: transition matrix, classification of states and stationary distribution.
The mistakes that cost marks
Transition probability read as long-run proportion. A one-step transition probability and a stationary probability are different objects. Markov questions are built to test that distinction.
Rows of a transition matrix not summing to one. Every row is a probability distribution over next states. Checking this first catches most setup errors.
Generating functions treated as algebra. They encode a distribution. Knowing which derivative gives which moment is the point, not the manipulation.
Ignoring the 45% threshold. The dual pass rule is on the final examination alone. Planning around your total mark hides the real constraint.
Formula & concept sheet
The vocabulary and formulas you must own
- Probability axioms
- The three rules from which all probability results follow: non-negativity, total mass one, and additivity for disjoint events.
- Conditional probability
- The probability of an event given that another has occurred, and the basis of Bayes' theorem.
- Random variable
- A function assigning a number to each outcome, described by a mass function or a density function.
- Expectation
- The probability-weighted average value of a random variable.
- Variance
- The expected squared deviation from the mean, measuring spread.
- Covariance
- A measure of joint variability; zero for independent variables, but zero covariance does not imply independence.
- Probability generating function
- A function encoding a discrete distribution, whose derivatives at one yield the moments.
- Law of large numbers
- The result that the sample mean converges to the true mean as the sample grows.
- Central limit theorem
- The result that sums of independent variables are approximately normal regardless of the underlying distribution.
- Markov property
- The condition that the next state depends only on the current state, not on the path taken to reach it.
- Transition matrix
- The array of one-step transition probabilities, each row summing to one.
- Stationary distribution
- The long-run proportion of time a chain spends in each state, unchanged by further transitions.
Common acronyms: CDF · CLT · LLN · PGF · PMF.
Where it fits
Prerequisites, related courses & why it matters
Prerequisite published by the department: 15 points from MATHS 108, 110, 130, 150 or 153, or equivalent. STATS 125 is worth 15 points and is taught in Semester One at the City campus. It is the standard entry point to the Stage II and III probability and stochastic processes courses.
Your STATS125 study toolkit
Study the course with Sia, not just read about it
Each tool already knows STATS125: 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 125 hard?
It rates moderately hard, and it is a genuine step up from Stage I applied statistics. The material is mathematical rather than data-driven, and the dual pass rule means the final examination must be passed on its own terms.
What is the assessment breakdown?
Final examination 50%, test 20%, assignments 18%, tutorials 12%. You must obtain at least 50% overall and at least 45% in the final examination alone.
How is it different from STATS 101?
Almost entirely. STATS 101 teaches reasoning with data and deliberately avoids formula manipulation. STATS 125 develops probability as mathematics — generating functions, limit theorems, Markov chains — and assumes a calculus prerequisite.
What do I need before taking it?
15 points from MATHS 108, 110, 130, 150 or 153, or equivalent. The calculus matters: continuous distributions require integration.
Are tutorials really graded?
Yes, at 12%. That is unusual and worth knowing early — it makes weekly attendance and preparation directly worth marks rather than merely advisable.
What is the hardest part?
Usually probability generating functions, because they are the first genuinely abstract tool in the course, followed by classifying states in a Markov chain. Both reward practice over reading.
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