Statistics Tuition Singapore | SG Maths Tuition

Statistics & Probability

I'm happy to help across a broad range of topics in statistics and probability. That can mean the descriptive basics, the inferential ideas behind them, or applied work in regression, hypothesis testing and machine learning.

Probability, Bayes' Theorem, discrete and continuous random variables, common distributions, expectation, variance, covariance, correlation
Multivariate distributions, marginal distributions and densities, conditional distributions and expectations, linear combinations of random variables
The law of large numbers, central limit theorem, sampling distributions, confidence intervals, hypothesis testing, regression analysis, convergence of random variables, statistical power
Goodness of fit tests, tests of association, non-parametric tests, bootstrapping, resampling techniques
Likelihood, maximum likelihood estimation (MLE), sufficiency, optimal tests of hypotheses, ANOVA, non-parametric and robust statistics
Bayesian statistics and Bayesian inference, Markov chain Monte Carlo (MCMC), hierarchical models

Where statistics and probability usually trip students up

Mathematical statistics catches people out in two directions at once: the probability is more subtle than it looks, and the calculus underneath is heavier than expected.

Probability intuition versus the formal rules. Conditional probability and Bayes' theorem are where good guesses go wrong, so the working has to lead, not the gut feeling.
The calculus demand. Continuous distributions mean integrating densities, and joint distributions mean double integrals, which surprises students who came for the statistics.
Population versus sample. Blurring a parameter with its estimate, or an estimator with a single estimate, undermines everything that follows in inference.
Reading confidence intervals and p-values. These are the most misquoted ideas in the subject, and saying precisely what they do and do not mean matters for the marks.
Choosing the right procedure. Picking the appropriate test or interval for the data in front of you is often harder than the calculation itself.
The step up to mathematical statistics. Proving properties of estimators is a different game from applying formulae, and it takes a more careful style of argument.

Before you start: a statistics prerequisites checklist

Mathematical statistics leans on more calculus than most people expect, so the prerequisites matter here. Before you start, it is worth confirming the following are solid.

What to be comfortable with before week one
Be comfortable with	Why it matters for statistics
Single-variable calculus	Differentiating and integrating confidently is needed for continuous distributions and for maximum likelihood.
Multivariable calculus	Double integrals appear as soon as joint and marginal distributions are introduced.
Basic combinatorics	Counting outcomes underpins discrete probability and many distribution derivations.
Some linear algebra	Vectors and matrices make regression and multivariate work far easier to follow.
Summation and notation	Sums, products and subscripted variables are used on almost every line.
Algebraic fluency	Rearranging expressions cleanly keeps long derivations, such as a likelihood, from going astray.

A one-semester self-study plan for independent learners

Statistics is one of the most widely useful subjects to teach yourself, whether for research, data work or another quantitative degree. If you are working through it alone, here is a thirteen-week route map built around texts at a genuine mathematical level. Work at the pace that suits you, and get in touch if estimation or hypothesis testing is not yet settling, since those ideas anchor everything applied that comes later.

Texts I often point self-study students to
Text	Author	Best for
A First Course in Probability	Sheldon Ross	A clear, example-led grounding in the probability half
Mathematical Statistics with Applications	Wackerly, Mendenhall & Scheaffer	A well-paced standard course text bridging probability and inference
Introduction to Mathematical Statistics	Hogg, McKean & Craig	A rigorous treatment of estimation and testing
Statistical Inference	Casella & Berger	The graduate standard for inference, for going deeper
Bayesian Data Analysis	Gelman and colleagues	A reference for the Bayesian material at the end

A thirteen-week plan
Week	Focus	Suggested reading	Checkpoint
1	Probability, conditioning and Bayes	Ross, opening chapters	Apply conditional probability and Bayes' theorem correctly
2	Random variables and discrete distributions	Ross or Wackerly	Work with the common discrete distributions
3	Continuous distributions, expectation and variance	Ross or Wackerly	Integrate densities to find probabilities and moments
4	Joint distributions and covariance	Wackerly	Handle marginal and conditional distributions
5	Functions of random variables and moment generating functions	Wackerly or Hogg	Find the distribution of a transformed variable
6	Law of large numbers and the central limit theorem	Ross or Wackerly	State and use the central limit theorem
7	Sampling distributions	Wackerly or Hogg	Derive the distribution of common sample statistics
8	Point estimation and maximum likelihood	Hogg or Wackerly	Find a maximum likelihood estimator and check it
9	Properties of estimators	Hogg or Casella & Berger	Assess bias, consistency and sufficiency
10	Confidence intervals	Wackerly or Hogg	Construct and interpret a confidence interval precisely
11	Hypothesis testing	Hogg or Casella & Berger	Set up a test and read a p-value correctly
12	Linear regression and ANOVA	Wackerly	Fit and interpret a simple linear model
13	Introduction to Bayesian inference and review	Gelman and colleagues, opening chapters	Contrast the Bayesian and frequentist views

Frequently Asked Questions (FAQ) - Statistics & Probability Tuition

Which statistics and probability topics and modules do you cover?

Probability and random variables, joint and multivariate distributions, the central limit theorem and sampling distributions, estimation and maximum likelihood, hypothesis testing, regression and ANOVA, and an introduction to Bayesian inference. That covers the probability and mathematical-statistics modules taken at NUS, NTU, SMU and the other local universities. Send me your module outline and I will tell you what I would focus on.

I did not expect this much calculus. Can you help me keep up?

Yes, and this catches many students out. Continuous probability needs integration, and joint distributions need double integrals, so part of my work is shoring up exactly the calculus your statistics module assumes, alongside the statistics itself. I patch the gap rather than letting it slow everything down.

Can you help with the statistics inside a data-science, economics or psychology module?

Often, yes. A great deal of applied statistics, such as regression, hypothesis testing and resampling, appears inside modules with other names. Tell me which module it is and where you are getting stuck, and I will say honestly whether it is a good fit.

Do you cover Bayesian statistics and computational methods too?

Yes. Bayesian inference, hierarchical models, and the idea behind Markov chain Monte Carlo are all part of the work I teach, and they sit on this same page rather than a separate one.

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