Probability & Statistics for Data Science

1️⃣ Probability Foundations

1.1: Understand Random Variables & Sample Spaces

1. Learn the difference between discrete and continuous random variables.
2. Understand sample spaces, events, and event operations (union, intersection, complement).
3. Grasp the basics of probability axioms and Kolmogorov’s rules.

Deeper Insight: Expect questions like, “If two events are independent, what does that mean intuitively and mathematically?” or “Can mutually exclusive events be independent?”

1.2: Conditional Probability & Independence

1. Understand conditional probability: [ P(A|B) = \frac{P(A \cap B)}{P(B)} ]
2. Grasp independence and conditional independence — crucial for ML concepts like Naïve Bayes or Bayesian Networks.
3. Learn chain rule of probability and total probability theorem.

Probing Question: “Why is conditional independence the cornerstone of graphical models?” or “How do you compute joint probabilities for dependent events?”

1.3: Bayes’ Theorem & Bayesian Reasoning

1. Derive Bayes’ theorem: [ P(A|B) = \frac{P(B|A)P(A)}{P(B)} ]
2. Understand priors, likelihoods, and posteriors.
3. Study Bayesian vs Frequentist interpretations — be ready to defend both approaches.

Deeper Insight: In interviews, they often link Bayes’ theorem to spam detection or A/B test updates — “How would you update your belief after new data?”

1.4: Combinatorics & Counting

1. Learn permutations, combinations, and binomial coefficients.
2. Understand sampling with/without replacement — a favorite in probability puzzles.

Probing Question: “If you pick 3 cards from a deck, what’s the probability that they’re all face cards?” This tests your combinatorial reasoning speed and clarity.

2️⃣ Probability Distributions

2.1: Core Discrete Distributions

1. Learn Bernoulli, Binomial, Poisson, and Geometric distributions.
2. For each, know their PMF, expected value, and variance.
3. Understand their use cases (e.g., Binomial → success/failure trials).

Probing Question: “If a rare event happens twice in 1000 trials, what distribution models it?”

2.2: Core Continuous Distributions

1. Learn Uniform, Normal (Gaussian), Exponential, and Gamma distributions.
2. Understand PDF and CDF relationships.
3. Study Standardization (Z-scores) and properties of Normal distributions.

Deeper Insight: Interviewers love “CLT intuition” — why sums of random variables tend toward Gaussian.

2.3: Joint, Marginal, and Conditional Distributions

1. Define and manipulate joint distributions ( P(X, Y) ).
2. Compute marginals and conditionals.
3. Learn Covariance and Correlation, and why “zero correlation” ≠ “independence.”

Probing Question: “Given Cov(X, Y) = 0, are X and Y independent?”

3️⃣ Statistical Inference & Estimation

3.1: Sampling & Estimation

1. Learn sample mean, variance, standard error, and sampling distributions.
2. Study Law of Large Numbers (LLN) and Central Limit Theorem (CLT).
3. Understand point vs interval estimation.

Deeper Insight: The CLT often appears as: “Why is normality assumed in so many models?”

3.2: Maximum Likelihood Estimation (MLE)

1. Derive MLE for simple distributions (Bernoulli, Gaussian).
2. Understand the intuition — choosing parameters that maximize the likelihood of observed data.
3. Learn numerical optimization for MLE (e.g., gradient-based methods).

Probing Question: “What if the likelihood is non-convex? How would you ensure convergence?”

3.3: Hypothesis Testing

1. Learn the null and alternative hypothesis framework.
2. Understand p-values, significance levels, Type I/II errors.
3. Practice with z-tests, t-tests, chi-square, ANOVA.

Deeper Insight: Expect A/B testing questions — “What’s your decision if the p-value is 0.06?”

3.4: Confidence Intervals

1. Learn how to build confidence intervals for means and proportions.
2. Connect confidence levels (95%) with sampling variability.
3. Understand bootstrapping as a non-parametric alternative.

Probing Question: “If your confidence interval includes zero, what does it mean for your hypothesis?”

4️⃣ Correlation, Regression & Association

4.1: Covariance, Correlation, and Their Pitfalls

1. Learn formulas for covariance and correlation.
2. Understand spurious correlation and Simpson’s paradox.

Probing Question: “Why doesn’t correlation imply causation? Give an example from data science.”

4.2: Simple Linear Regression (Statistical View)

1. Derive regression coefficients via least squares.
2. Understand residuals, R², and assumptions (linearity, normality, independence).
3. Contrast statistical regression vs machine learning regression.

Deeper Insight: Be ready to discuss how “violating assumptions” (e.g., heteroscedasticity) affects reliability.

5️⃣ Advanced Topics for Data Science Interviews

5.1: Bayesian Inference & Priors

1. Learn conjugate priors (e.g., Beta-Binomial, Normal-Normal).
2. Understand posterior predictive distributions.
3. Practice simple Bayesian updates by hand.

Probing Question: “If you have little data, how does your choice of prior affect the posterior?”

5.2: Resampling & Validation

1. Understand bootstrap and jackknife methods.
2. Connect these to cross-validation in ML.
3. Learn bias-variance tradeoff mathematically.

Deeper Insight: “Why might bootstrap overestimate model variance on small datasets?”

5.3: Experimental Design

1. Learn randomization, control, and blocking.
2. Study A/B/n testing, sequential testing, and false discovery correction.
3. Connect to causal inference (randomization ⇒ independence).

Probing Question: “What if your A/B test shows a lift, but you suspect Simpson’s paradox?”