**GATE Statistics Syllabus:**

Calculus:
Finite, countable and uncountable sets, Real number system as a complete
ordered field, Archimedean
property; Sequences and series, convergence; Limits, continuity, uniform
continuity, differentiability, mean value theorems; Riemann integration,
Improper integrals; Functions of two or three variables, continuity,
directional derivatives, partial derivatives, total derivative, maxima and
minima, saddle point, method of Lagrange’s multipliers; Double and Triple
integrals and their applications; Line integrals and Surface integrals, Green’s
theorem, Stokes’ theorem, and Gauss divergence theorem.

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Linear Algebra:
Finite dimensional vector spaces over real or complex fields; Linear
transformations and their matrix representations, rank; systems of linear equations,
eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem,
diagonalization, Jordan canonical form, symmetric, skew-symmetric, Hermitian,
skew- Hermitian, orthogonal and unitary matrices; Finite dimensional inner
product spaces, Gram-Schmidt orthonormalization process, definite forms.

Probability:
Classical, relative frequency and axiomatic definitions of probability,
conditional probability, Bayes’ theorem, independent events; Random variables and
probability distributions, moments and moment generating functions, quantiles;
Standard discrete and continuous univariate distributions; Probability
inequalities (Chebyshev, Markov, Jensen); Function of a random variable;
Jointly distributed random variables, marginal and conditional distributions,
product moments, joint moment generating functions, independence of random
variables; Transformations of random variables, sampling distributions,
distribution of order statistics and range; Characteristic functions; Modes of
convergence; Weak and strong laws of large numbers; Central limit theorem for
i.i.d. random variables with existence of higher ordermoments.

Stochastic Processes: Markov chains with finite and countable state space, classification
of states, limiting behaviour of n-step transition probabilities, stationary
distribution, Poisson and birth-and-deathprocesses.

Inference:
Unbiasedness, consistency, sufficiency, completeness, uniformly minimum
variance unbiased estimation, method of moments and maximum likelihood estimations;
Confidence intervals; Tests of hypotheses, most powerful and uniformly most
powerful tests, likelihood ratio tests, large sample test, Sign test, Wilcoxon signed rank
test, Mann- Whitney U test, test for independence and Chi-square test for
goodness of fit.

Regression Analysis: Simple and multiple linear regression, polynomial regression,
estimation, confidence intervals and testing for regression coefficients; Partial and
multiple correlationcoefficients.

Multivariate Analysis: Basic properties of multivariate normal distribution;
Multinomial distribution; Wishart distribution; Hotellings T

^{2}and related tests; Principal component analysis; Discriminant analysis; Clustering.^{2}and 2

^{3}Factorial experiments.

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