Mathematics and Statistics Foundations
What to know cold
Expect questions that test whether you can pick the right tool and reason about its assumptions, not just compute. Core topics: probability (conditional probability, Bayes' theorem, independence vs. mutual exclusivity), random variables and their expectations/variances, common distributions (Bernoulli, binomial, Poisson, Gaussian), sampling distributions and the central limit theorem, estimation (bias, variance, consistency, MLE), hypothesis testing (p-values, Type I/II errors, power), confidence intervals, covariance/correlation, and linear algebra (vectors, matrices, eigenvalues, projections, rank, positive definiteness). Calculus appears mainly through gradients and convexity.
Traps
- Confusing P(A|B) with P(B|A) - always write Bayes' theorem out.
- Treating "independent" and "mutually exclusive" as the same; disjoint events with nonzero probability are dependent.
- Misreading a p-value as the probability the null is true.
- Forgetting that correlation measures only linear association and is scale-invariant.
- Assuming the CLT applies to the raw data rather than to the sampling distribution of a mean.
Rule of thumb
Variance of a sum: Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y); the covariance term vanishes only under independence.
Worked example
A test has 1% prevalence, 99% sensitivity, 95% specificity. P(disease|positive) = (0.99·0.01)/(0.99·0.01 + 0.05·0.99) ≈ 0.0099/0.0594 ≈ 0.167. Low base rates crush posterior probability - a classic trap.