ST covers calculus, matrix theory, probability, distributions, convergence, stochastic processes, estimation, testing, non-parametric methods, multivariate analysis and regression.

Statistics is tested within a 100-mark paper. The paper also includes General Aptitude. The sections below show the complete subject syllabus.

GATE ST Syllabus and Marks Distribution

There is no optional subject block in this paper. All listed sections are part of the syllabus.

Statistics Unit-wise Syllabus

Calculus

• Finite, countable and uncountable sets; Sequences of real numbers, convergence of sequences, bounded sequences, monotonic sequences, Cauchy criterion for convergence; Series of real numbers, convergence, tests of convergence, alternating series, absolute and conditional convergence; Power series and radius of convergence; Functions of a real variable: Limit, continuity, monotone functions, uniform continuity, differentiability, Rolle’s theorem, mean value theorems, Taylor’s theorem, L’Hospital rules, maxima and minima, Riemann integration and its properties, improper integrals;
• Functions of several real variables: Limit, continuity, partial derivatives, directional derivatives, gradient, Taylor’s theorem, total derivative, maxima and minima, saddle point, method of Lagrange multipliers, double and triple integrals and their applications.

Matrix Theory:

• Subspaces of Rn and Cn, span, linear independence, basis and dimension, row space and column space of a matrix, rank and nullity, row reduced echelon form, trace and determinant, inverse of a matrix, systems of linear equations; Inner products in R n and Cn, Gram-Schmidt orthonormalization; Eigen values and eigen vectors, characteristic polynomial, Cayley-Hamilton theorem, symmetric, skew-symmetric, Hermitian, skew- Hermitian, orthogonal, unitary matrices and their eigen values, change of basis matrix, equivalence and similarity, diagonalizability, positive definite and positive semi-definite matrices and their properties, quadratic forms, singular value decomposition.

Probability

• Axiomatic definition of probability, properties of probability function, conditional probability, Bayes’ theorem, independence of events; Random variables and their distributions, cumulative distribution function, probability mass function, probability density function and their properties, expectation, variance, properties of expectation and variance, moments and moment generating function, probability generating function, quantiles, distribution of functions of a random variable, Chebyshev, Markov and Jensen inequalities.

Standard Univariate Distributions

• Bernoulli, binomial, geometric, negative binomial, hypergeometric, discrete uniform, Poisson, continuous uniform, exponential, double exponential, gamma, beta distributions of first and second kinds, Weibull, normal, Cauchy.

Joint Distributions

• Jointly distributed random variables and their distribution functions, probability mass function, probability density function and their properties, marginal and conditional distributions, conditional expectation and moments, product moments, simple correlation coefficient, joint moment generating function, independence of random variables, functions of random vector and their distributions, distributions of order statistics, joint and marginal distributions of order statistics; multinomial distribution, bi variate normal distribution, sampling distributions: central chi-square, central t, and central F distributions.

Convergence of Random Variables

• Convergence in distribution, convergence in probability, convergence almost surely, convergence in r − th mean and their inter -relations, Slutsky’s lemma, Borel -Cantelli lemma; weak and strong laws of large numbers; central limit theorem for i.i.d. random variables.

Stochastic Processes

• Markov chains with finite and countable state space, classification of states, limiting behaviour of n -step transition probabilities, stationary distribution, Poisson process, birth-and-death process, pure-birth process, pure-death process, Brownian motion and its basic properties.

Estimation

• Sufficiency, minimal sufficiency, factorization theorem, completeness, completeness of exponential families, ancillary statistic, Basu’s theorem and its applications, unbiased estimation, uniformly minimum variance unbiased estimation, Rao-Blackwell theorem, Lehmann-Scheffe theorem, Cramer-Rao inequality, consistent estimators, method of moments estimators, method of maximum likelihood estimators and their properties;
• Interval estimation: pivotal quantities and confidence intervals based on them, coverage probability.

Testing of Hypotheses

• Neyman-Pearson lemma, most powerful tests, monotone likelihood ratio (MLR) property, uniformly most powerful tests, uniformly most powerful tests for families having MLR property, uniformly most powerful unbiased tests, uniformly most powerful unbiased tests for exponential families, likelihood ratio tests, large sample tests.

Non-parametric Statistics

• Empirical distribution function and its properties, goodness of fit tests, chi-square test, Kolmogorov-Smirnov test, run tests, sign test, Wilcoxon signed rank test, Mann -Whitney U -test, Kruskal -Wallis test, rank correlation coefficients of Spearman and Kendall.

Multivariate Analysis

• Multivariate normal distribution: properties, conditional and marginal distributions, maximum likelihood estimation of mean vector and dispersion matrix, Hotelling’s T2 test, Wishart distribution and its basic properties, multiple and partial correlation coefficients and their basic properties.

Regression Analysis

• Simple and multiple linear regression, R2 and adjusted R2 and their applications, distributions of quadratic forms of random vectors: Fisher-Cochran theorem, Gauss-Markov theorem, tests for regression coefficients, confidence intervals.

General Aptitude Syllabus

General Aptitude contributes 15 marks within the 100-mark paper. It is not an additional paper that can be left out when choosing the technical sections.

Chapter-wise Weightage and Important Topics

Only the section-wise marks are fixed. IIT Madras has not assigned a guaranteed number of marks to each chapter of Statistics. A topic may appear in one year and receive fewer questions in another; a past-paper count is not an official 2027 weightage.

For a chapter-weightage comparison, record the year, paper code, session, question number and marks of each matching question. Keep a combined question under one main topic or label the overlap explicitly; counting it twice inflates the total. GATE previous-year question papers can be used for this topic-by-topic practice.

Question Types and Negative Marking

The paper uses one-mark and two-mark questions. The number of MCQ, MSQ and NAT questions is not a fixed chapter-wise quota. In an MSQ, selecting one correct option is not enough when the correct answer contains two options.