Covers fundamental probability theory and statistical inference from first principles to practical applications. Core probability concepts include sample spaces and events, independence, conditional probability, Bayes theorem, expected value, variance, and standard deviation. Reviews common probability distributions such as normal, binomial, Poisson, uniform, and exponential, their parameters, typical use cases, computation of probabilities, and approximation methods. Explains sampling distributions and the Central Limit Theorem and their implications for estimation and confidence intervals. Presents descriptive statistics and data summary measures including mean, median, variance, and standard deviation. Details the hypothesis testing workflow including null and alternative hypotheses, p values, statistical significance, type one and type two errors, power, effect size, and interpretation of results. Reviews commonly used tests and methods and guidance for selection and assumptions checking, including z tests, t tests, chi square tests, analysis of variance, and basic nonparametric alternatives. Emphasizes practical issues such as correlation versus causation, impact of sample size and data quality, assumptions validation, reasoning about rare events and tail risks, and communicating uncertainty. At more advanced levels expect experimental design and interpretation at scale including A B tests, sample size and power calculations, multiple testing and false discovery rate adjustment, and design choices for robust inference in real world systems.
EasyTechnical
94 practiced
Explain Type I and Type II errors in hypothesis testing. Give an ML example where minimizing Type II error is more important than minimizing Type I error, and explain why.
Sample Answer
Type I error (false positive): rejecting a true null hypothesis. Type II error (false negative): failing to reject a false null. Example where minimizing Type II is more important: fraud detection in high-value transactions—missing a fraud (Type II) can cause large monetary loss and reputational damage, while a false alarm (Type I) may be tolerable because it triggers manual review. In ML, we trade off these via thresholds, loss weighting, or different operating points on ROC: we might lower threshold to reduce Type II at cost of more Type I, and monitor human review costs and customer experience.
EasyTechnical
59 practiced
Define the Central Limit Theorem (CLT) and explain its practical implication for constructing confidence intervals for the mean when n is large. When does CLT not apply?
Sample Answer
Central Limit Theorem: for i.i.d. random variables with finite mean μ and variance σ^2, the sampling distribution of the sample mean (properly normalized) approaches a Normal(0,1) as n→∞. Practical implication: when n is large (commonly n≥30), we can approximate the sampling distribution of the mean as normal and construct confidence intervals: mean ± z_{α/2} * (s/√n). CLT fails or is unreliable when: samples are not independent, the distribution has infinite variance (heavy tails), or n is small and the underlying distribution is highly skewed—then use bootstrap or exact small-sample methods (t-distribution if normality plausible). For ML, CLT justifies use of normal-based uncertainty estimates for aggregated metrics in A/B tests with large traffic.
HardTechnical
81 practiced
Derive the sampling distribution of the sample mean for i.i.d. random variables with mean μ and variance σ^2. Show the mean and variance of the sampling distribution and explain implications for estimator consistency.
Sample Answer
Let X1,...,Xn be i.i.d. with mean μ and variance σ^2. The sample mean X̄ = (1/n) Σ Xi.By linearity: E[X̄] = (1/n) Σ E[Xi] = μ — so X̄ is unbiased.Variance: Var(X̄) = (1/n^2) Σ Var(Xi) = n·σ^2 / n^2 = σ^2 / n. Thus as n increases, variance shrinks at rate 1/n.If Xi have finite variance, by the Central Limit Theorem, for large n: (X̄ - μ) / (σ/√n) → N(0,1). So sampling distribution approximates normal centered at μ with variance σ^2/n.Implications for consistency: since Var(X̄) → 0 and E[X̄]=μ, X̄ converges in probability to μ (by Chebyshev or law of large numbers), hence X̄ is a consistent estimator. Practically this means increasing sample size improves estimator precision and justifies normal-based confidence intervals when n large.
HardTechnical
62 practiced
Discuss Bayesian vs frequentist approaches for estimating a conversion rate in an online experiment. Provide priors you might choose for a Bayesian model, explain how posterior intervals differ from confidence intervals, and describe a scenario where Bayesian methods are preferable in ML engineering.
Sample Answer
Situation: Estimating conversion rate p in online experiments can follow frequentist or Bayesian paradigms.Frequentist approach: model conversions as Binomial(n,p); use MLE p̂ = k/n, construct Wald or Wilson confidence intervals. Good when n large and you want long-run frequency guarantees.Bayesian approach: place prior on p (Beta). Priors: Beta(1,1) uniform for noninformative; Beta(α,β) informed by historical CTR (e.g., α=historical_successes+1, β=historical_failures+1); Beta(0.5,0.5) Jeffreys prior for objective inference.Posterior intervals vs confidence intervals: a Bayesian credible interval gives probability p lies in interval conditional on data and prior (P(p in I | data)=95%). A 95% frequentist confidence interval means that over repeated experiments 95% of such intervals will contain true p — not a direct probability statement about p.When Bayesian preferable in ML engineering: small-sample or sparse-signal regimes (new feature with low traffic), hierarchical pooling across segments (partial pooling to stabilize estimates), and when you need direct probabilistic statements for downstream decision-making or sequential experimentation with stopping rules — Bayesian updating naturally supports continual monitoring and decision thresholds (e.g., P(p_A>p_B)>0.99) while coherently incorporating prior knowledge and uncertainty.
EasyTechnical
47 practiced
Explain what a p-value is and what it is not. Provide a concise interpretation for a p-value of 0.03 in the context of testing whether a new recommendation model increases click-through rate compared to baseline.
Sample Answer
A p-value is the probability, under the null hypothesis, of observing data as extreme or more extreme than what was observed. It is not the probability that the null is true, nor the size of the effect. Interpretation for p=0.03 in testing whether a new recommender increases CTR: assuming the null (no increase) is true, there's a 3% chance we'd observe the measured uplift (or larger) due to random variation. This suggests evidence against the null at the 5% level, but you should also report effect size, confidence interval, potential multiple-testing, and practical significance before deploying the model.
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