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# Topics in Entrance Examination Graduate Programs, Department of Mathematics

**Mathematical Statistics**

Random variables and their distribution functions; probability density and frequency functions; mathematical expectations and moments; moment generating functions; characteristic functions; probability inequalities; special discrete and continuous distributions; multivariate random vectors and their distributions; conditional and marginal distributions; conditional expectation; transformation and functions of random variables; modes of convergence; Central Limit Theorem; sampling distribution; estimation and testing of statistical hypotheses; principles of estimation: unbiasedness, minimum variance and consistency; sufficiency; multi-parameter estimation; maximum likelihood and moment estimates.

**Linear Algebra**

Systems of linear equations; matrices & determinants; eigenvalues; eigenvectors; canonical forms; vector spaces; linear independence; bases; orthonormal bases; linear transformations; ranges; kernels; ranks; nullities; isomorphisms; solution spaces of linear differential equations.

**Advanced Calculus**

Sequences; series; convergence tests; limits; continuity; derivatives; integrals; Fundamental Theorems of Calculus; Intermediate Value Theorem; Mean Value Theorem; Taylor series expansion; curvilinear coordinates; change of coordinates; partial derivatives; multiple integrals and change of variables.

Complex Variables

Complex numbers; analytic functions; Cauchy-Riemann equations; conformality; analytic continuation; Cauchy's Theorems; maximum modulus principle; Liouville's Theorem; Residue Theorems and evaluation of real integrals; principle of argument; Rouche's Theorem.