MATHEMATICS

  • Sequences and Series:

    Convergence of sequences of real numbers, Comparison, root and ratiotests for convergence of series of real numbers.

  • Differential Calculus:

    Limits, continuity and differentiability of functions of one and two variables.Rolle's theorem, mean value theorems, Taylor's theorem, indeterminate forms, maxima and minimaof functions of one and two variables.

  • Integral Calculus:

    Fundamental theorems of integral calculus. Double and triple integrals, applicationsof definite integrals, arc lengths, areas and volumes.

  • Matrices:

    Rank, inverse of a matrix. Systems of linear equations. Linear transformations, eigenvaluesand eigenvectors. Cayley-Hamilton theorem, symmetric, skew-symmetric and orthogonal matrices.

    Differential Equations:

    Ordinary differential equations of the first order of the form y'=f(x,y), Bernoulli's equation, exact differential equations, integrating factor, orthogonal trajectories, homogeneous differential equations, variable separable equations, linear differential equations of second order with constant coefficients, method of variation of parameters, Cauchy-Euler equation.

    STATISTICS

  • Probability:

    Axiomatic definition of probability and properties, conditional probability, multiplication rule. Theorem of total probability. Bayes' theorem and independence of events.

  • Random Variables:

    Probability mass function, probability density function and cumulative distributionfunctions, distribution of a function of a random variable. Mathematical expectation, moments andmoment generating function. Chebyshev's inequality.

  • Standard Distributions:

    Binomial, negative binomial, geometric, Poisson, hypergeometric, uniform, exponential, gamma, beta and normal distributions. Poisson and normal approximations of a binomial distribution.

  • Sampling distributions:

    Chi-square, t and F distributions, and their properties.

  • Limit Theorems:

    Weak law of large numbers. Central limit theorem (i.i.d.with finite variance case only).

  • Estimation:

    Unbiasedness, consistency and efficiency of estimators, method of moments and method of maximum likelihood. Sufficiency, factorization theorem. Completeness, Rao-Blackwell and Lehmann-Scheffe theorems, uniformly minimum variance unbiased estimators. Rao-Cramer inequality. Confidenceintervals for the parameters of univariate normal, two independent normal, and one parameter exponentialdistributions.

  • Testing of Hypotheses:

    Basic concepts, applications of Neyman-Pearson Lemma for testing simpleand composite hypotheses. Likelihood ratio tests for parameters of univariate normal distribution.