Probability and Statistics
“Master counting combinatorics, discrete/continuous distributions, bivariate joint PDFs, Central Limit Theorem, MLE parameter estimation, and hypothesis testing for GATE DA.”
A comprehensive academic study notebook covering counting principles, probability axioms, Bayes' Theorem, classical discrete and continuous distributions, joint and conditional distributions, covariance matrices, sampling distributions (Chi-square, t, F), Central Limit Theorem, point estimation (MLE), and statistical hypothesis testing (z, t, Chi-square). Modeled on the reference Operating Systems benchmark for the official GATE Data Science & AI (DA) curriculum.
Structured Notes & Exam Lens
A · B.Tech Structured Study Sequence
5 Sections · Full CurriculumProbability Axioms, Counting & Bayes' Theorem
Combinatorics, Kolmogorov's probability axioms, independent and mutually exclusive events, conditional probability, Law of Total Probability, and Bayesian posterior updating.
Permutations, combinations, Kolmogorov axioms, and independent vs mutually exclusive events.
Partitions of sample space, marginalization, prior and posterior probabilities, and likelihood ratios.
Discrete & Continuous Distributions
Random variables, Probability Mass Functions (PMF), Probability Density Functions (PDF), Cumulative Distribution Functions (CDF), Binomial, Poisson, Uniform, Exponential, and Gaussian distributions.
Expectation E[X], Variance Var(X), Bernoulli trials, Binomial PMF, and Poisson limit theorem.
PDF integration, CDF properties, Continuous Uniform, Exponential distribution, and the memoryless property.
Symmetric bell curve, inflection points at mu ± sigma, Z-score transformation, and table integration.
Joint Distributions, Covariance & Limit Theorems
Bivariate random variables, marginalization, conditional expectation and variance, covariance matrices, Chebyshev's inequality, Law of Large Numbers, and the Central Limit Theorem.
Double integrals over coupled domains, marginal PDFs, conditional density, and iterated expectation.
Covariance Cov(X, Y), Pearson correlation rho, Positive Semi-Definite covariance matrices, and bivariate Gaussian independence.
Markov bound, Chebyshev tail bound, Weak Law of Large Numbers, and Gaussian convergence of sample sums.
Sampling Distributions & Maximum Likelihood Estimation
Sampling distributions (Chi-square, Student's t, Snedecor's F), point estimation properties (bias, consistency, efficiency), and Maximum Likelihood Estimation (MLE).
Degrees of freedom, ratios of Chi-squares, heavy tails, and small-sample statistics.
Likelihood functions, log-likelihood optimization, score function, Hessian concavity, and estimator properties.
Confidence Intervals & Hypothesis Testing
Confidence intervals, Null and Alternative hypotheses, Type I and Type II errors, p-values, z-test, Student's t-test, and Chi-square goodness-of-fit / contingency tests.
Constructing intervals, rejection regions, p-value decision rules, and two-sample statistical tests.
B · GATE 2027 Syllabus Scope
Official syllabus scope and related lesson coverage; topics outside the syllabus are marked as enrichment.
- •Counting, permutations, combinations, probability axioms, sample space, independent events, mutually exclusive events
- •Marginal, conditional and joint probability, Bayes Theorem, conditional expectation and variance
- •Mean, median, mode, standard deviation, correlation, covariance, random variables, PMF, PDF, CDF
- •Distributions: Bernoulli, Binomial, Poisson, Uniform, Exponential, Gaussian/Normal, Student's t, Chi-square
- •Two-dimensional random variables, marginal and conditional distributions, covariance matrix
- •Central Limit Theorem (CLT), Law of Large Numbers, Chebyshev's inequality
- •Estimation & Inference: Maximum Likelihood Estimation (MLE), confidence intervals, z-test, t-test, Chi-square test
Interactive Labs & Verified PYQs
Revision & Rapid Review Sheets
Authoritative Textbooks & Video Lectures
All notes in this notebook cross-reference standard academic literature. We do not pretend student notes substitute for classical texts — they guide you through them.
Curated lectures from verified educators (Gate Smashers, Neso Academy, Knowledge Gate, and MIT OCW) selected specifically for difficult concept visualizations.
These notes are open and maintained together.
Found a factual mistake? Have a clearer explanation or mental model? Created a useful diagram or worked out a tricky GATE question? Help make this notebook better for every student who opens it next.