Includes probability, statistical inference, regression analysis, and data interpretation, for careers in statistics and data analysis.

GATE Statistics

Official 2025 syllabus extracted from https://gate2025.iitr.ac.in/doc/2025/GATE%20_ST_2025_Syllabus.pdf

Calculus

Finite, countable and uncountable sets

Real number system as a complete ordered field

Archimedean property

Sequences of real numbers: convergence, bounded, monotonic, Cauchy criterion

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, 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, minima, saddle point

Method of Lagrange multipliers

Double and triple integrals and their applications

Matrix Theory

Subspaces of R^n and C^n

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 C^n

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

Distribution function

Probability mass function

Probability density function and their properties

Expectation, moments and moment generating function

Quantiles

Distribution of functions of a random variable

Chebyshev, Markov and Jensen inequalities

Standard Discrete and Continuous Univariate Distributions

Bernoulli

Binomial

Geometric

Negative binomial

Hypergeometric

Discrete uniform

Poisson

Continuous uniform

Exponential

Gamma

Beta

Weibull

Normal

Standard Distributions

Joint Distributions and Sampling Distributions

Jointly distributed random variables and their distribution functions

Probability mass function and probability density function

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

Bivariate normal distribution

Sampling distributions: central chi-square, central t, central F distributions

Joint Distributions

Limit Theorems

Convergence in distribution

Convergence in probability

Convergence almost surely

Convergence in r-th mean

Inter-relations between modes of convergence

Slutsky's lemma

Borel-Cantelli lemma

Weak and strong laws of large numbers

Central limit theorem for i.i.d. random variables

Delta method

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

Coverage probability

Testing of Hypotheses

Neyman-Pearson lemma

Most powerful tests

Monotone likelihood ratio (MLR) property

Uniformly most powerful tests

UMP tests for families having MLR property

Uniformly most powerful unbiased tests

UMP 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

Sign test

Wilcoxon signed rank test

Mann-Whitney U-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 T-squared test

Wishart distribution and its basic properties

Multiple and partial correlation coefficients and their basic properties

Regression Analysis

Simple linear regression

Multiple linear regression

R-squared and adjusted R-squared and their applications

Distributions of quadratic forms of random vectors: Fisher-Cochran theorem

Gauss-Markov theorem

Tests for regression coefficients

Confidence intervals

GATE Statistics Syllabus 2025, 2026 and more: Subjects, Topics, and What to Study