Fundamentals of Probability and Statistics
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Trading interviews test probability harder than almost any other subject, and they test it in a particular way: not whether you can state a theorem, but whether you can set a problem up correctly under time pressure and notice when an answer is impossible.
This is the full foundation, in the order the ideas depend on each other. Axioms, counting and conditional probability first, then Bayes, then random variables and the distributions worth knowing cold. The middle covers the central limit theorem and what it does and does not promise. Frequentist inference follows, including the difference between a confidence interval and the probability statement people mistake it for, and why a year of profitable trading can be entirely consistent with having no edge. It ends on Bayesian inference and regression.
Every lesson carries worked numbers and most carry something you can move. School-level algebra is the only prerequisite; the calculus that appears is explained where it appears.
What is Probability?Probability is the language traders use to price uncertainty, and the foundation every quant interview is built on.Free previewTheoretical vs Empirical ProbabilityModels give you the clean number, data gives you the real one, and the gap between them is where trading edge and trading blowups both live.Free previewThree Views of ProbabilityClassical, frequentist and Bayesian interpretations answer different questions, and knowing which one a problem is asking for saves you the whole argument.Free previewSample Space and EventsDefining the outcome set precisely is the step candidates skip, and it is the step that causes most wrong answers.Rules of ProbabilityThe axioms plus four working rules, and the two conditions people forget to check before applying them.Independence and ExpectationExpectation is what you price off, and linearity is the single most useful shortcut in interview probability.Variance and Standard DeviationExpectation tells you the price; variance tells you how wrong you can be, which is what actually determines your position size.Covariance and CorrelationHow two variables move together is the whole basis of hedging and diversification, and correlation only captures the linear part of it.Key Inequalities in ProbabilityMarkov, Chebyshev and Jensen bound what you cannot compute, which is most of what matters when the distribution is unknown.
Set Operations in ProbabilityUnion, intersection and complement, plus De Morgan's laws, which are what let you rewrite an awkward event into an easy one.Counting MethodsPermutations and combinations, and the one question to ask first: does order matter?Advanced Counting TechniquesMultinomials, stars and bars, Catalan numbers and recurrences: the four patterns that cover the counting problems plain combinations cannot.
What is a Random Variable?A random variable turns outcomes into numbers, which is what makes them addable, averageable and priceable.Discrete vs Continuous VariablesPMFs assign probability to points, PDFs assign density to intervals, and confusing density with probability is the classic mistake.Using the PDF and CDFWorking with densities and cumulative functions, including the inverse-CDF trick that underpins every Monte Carlo simulation.
Bernoulli and Binomial DistributionsOne trial and many, the distributions behind hit rates, fill counts and the binomial option pricing tree.Poisson and Geometric DistributionsCounting rare events in a window, and waiting for the first success. Both show up constantly in order-flow questions.Negative Binomial and HypergeometricWaiting for the r-th success, and sampling from a finite pool without replacement.
Uniform and Normal DistributionsThe simplest continuous distribution and the most important one, plus the sigma levels every trader quotes.Exponential, Gamma, and BetaWaiting times, sums of waiting times, and a distribution over probabilities themselves.Central Limit TheoremWhy averages become normal regardless of what they average, and the two conditions that decide whether you can rely on it.
Confidence IntervalsA range with a stated long-run coverage, and a precise meaning that is not the one most people give it.Hypothesis TestingA structured way to ask whether a result could plausibly be noise, and the asymmetry built into how it answers.p-values and Statistical DecisionsWhat a p-value measures, the three things it is routinely mistaken for, and why p-hacking is the default failure mode of strategy research.Type I and Type II ErrorsFalse positives and false negatives, the tradeoff between them, and why most strategy research is badly underpowered.Statistical vs Practical SignificanceDetectable and worth acting on are different questions, and transaction costs decide the second one.Advanced Frequentist ToolsANOVA, rank-based tests, multiple-comparison corrections, the bootstrap and maximum likelihood, and when each earns its place.
Law of the Unconscious StatisticianCompute the expectation of a function of X without ever finding the distribution of that function.Moment Generating FunctionsA single function that encodes every moment, turns sums into products, and identifies a distribution uniquely.Markov Chain ProbabilitySystems where only the present matters, and the steady-state calculation that answers most interview questions about them.
Bayesian PhilosophyTreating the unknown parameter as uncertain rather than fixed, and what that buys you when data is scarce.Bayesian Updating with Bayes' RuleBelief revision as a repeatable loop, with the order-independence property that makes sequential updating safe.Prior, Likelihood, PosteriorThe three pieces and how they interact, including why the posterior always sits between the prior and the data.Bayesian ComputationWhen the posterior has no closed form, sample from it: MCMC, Gibbs, and how to tell whether the sampler worked.
What is Regression?Modelling how one variable moves with another, which in trading is how a hedge ratio gets computed.Ordinary Least SquaresThe estimator behind almost every fitted line, why it uses squared errors, and what BLUE actually guarantees.Multiple Linear RegressionSeveral predictors at once, what 'holding the others constant' really means, and the multicollinearity that wrecks factor models.Regression DiagnosticsChecking the assumptions before trusting the output, and knowing which violations bias coefficients versus which only break the standard errors.Advanced Regression TechniquesRegularisation, non-linear fits and time-series methods, and the bias-variance trade that justifies deliberately biased estimators.
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