| Section |
Topics |
Areas Covered / Subtopics |
| Section 1 |
Sequences and Series of real numbers |
Sequences of real numbers, convergence and limits, Cauchy sequences, monotonic sequences, limits of standard sequences, limit superior and inferior, infinite series (convergence/divergence), convergence of series with non-negative terms, comparison test, limit comparison test, D’Alembert’s ratio test, Cauchy’s n-th root test, Cauchy’s condensation test, integral test, absolute convergence, Leibnitz’s test for alternating series, conditional convergence, power series and radius of convergence. |
| Section 2 |
Differential Calculus of one and two real variables, and Integral Calculus |
Differential Calculus (one variable): Limits, continuity, differentiability, properties of continuous/differentiable functions, Rolle’s theorem, Lagrange’s mean value theorem, higher-order derivatives, Leibniz’s rule, Taylor’s theorem (Lagrange’s and Cauchy’s form), Taylor/Maclaurin series, indeterminate forms, L’Hospital’s rule, maxima and minima (local/global), points of inflection.
Differential Calculus (two variables): Limits, continuity, differentiability, partial/total differentiation, successive differentiation, maxima/minima, Hessian matrix, saddle points, constrained optimisation (Lagrange multiplier).
Integral Calculus: Fundamental theorems, Leibniz’s rule, differentiation under integral sign, improper integrals, Beta and Gamma integrals, double integrals, change of order, transformation of variables, and applications (arc lengths, areas, and volumes). |
| Section 3 |
Matrices and Determinants |
Vector spaces (ℝn and ℂn), span, linear dependence/independence, dimension, basis, null space, algebra of matrices, symmetric/skew-symmetric, Hermitian/skew-Hermitian, orthogonal/unitary, idempotent/nilpotent matrices, determinants (properties, applications, transformations), determinant of product, singular/non-singular matrices, trace, adjoint, inverse, rank and nullity, row-rank, column-rank, rank theorems, row reduction, echelon forms, systems of equations (consistent/inconsistent), Cramer’s rule, characteristic roots and vectors, Cayley-Hamilton theorem, quadratic forms, definiteness of matrices (positive, negative, semi-definite). |
| Section 4 |
Descriptive Statistics and Probability |
Descriptive Statistics: Concepts of sample and population, types of data, tabular/graphical representation, measures of central tendency (mean, median, mode, etc.), measures of dispersion (range, variance, SD, etc.), moments, skewness, kurtosis, bivariate data, covariance, correlations (simple, partial, multiple), and Spearman’s rank correlation.
Probability: Random experiments, sample space, event algebra, definitions of probability (relative frequency/axiomatic), properties, addition theorem, geometric probability, Boole’s and Bonferroni’s inequalities, conditional probability, multiplication rule, total probability theorem, Bayes’ theorem, independence of events. |
| Section 5 |
Univariate Distributions |
Random variables, cumulative distribution function (cdf), discrete/continuous variables, pmf, pdf, distribution of functions (Jacobian method), expectation and moments, mean, median, mode, variance, coefficient of variation, quantiles, skewness, kurtosis, moment generating function (mgf), inequalities (Markov, Chebyshev). Distributions: Degenerate, Bernoulli, Binomial, Negative binomial, Geometric, Poisson, Hypergeometric, Uniform, Exponential, Double exponential, Gamma, Beta (I & II), Normal, Cauchy. |
| Section 6 |
Multivariate Distributions |
Random vectors, joint/marginal cdfs, pmfs, pdfs, conditional cdfs, conditional distributions, independence of random variables, transformations (Jacobian), expectations of functions of random vectors, joint moments, covariance, correlation, joint MGFs, conditional moments/expectations, additive properties (Binomial, Poisson, Gamma, Normal), multinomial distribution, and bivariate normal distribution (marginal/conditional distributions, properties). |
| Section 7 |
Limit Theorems |
Convergence (in probability, mean square, almost sure, in distribution), inter-relations of convergences, weak law of large numbers, strong law of large numbers, and central limit theorem (i.i.d., finite variance). |
| Section 8 |
Sampling Distributions |
Random sample, parameter, statistic, sampling distribution, order statistics, distributions of smallest/largest order statistics (discrete/continuous), Chi-square distribution (definition, pdf derivation, properties, additive property, limiting form), t-distribution (definition, pdf derivation, properties, limiting form), and F-distribution (definition, pdf derivation, properties, reciprocal distribution, relationships between t, F, χ²). |
| Section 9 |
Estimation |
Properties of estimators: unbiasedness, sufficiency, consistency, relative efficiency, complete statistic, UMVUE, Rao-Blackwell theorem, Lehmann-Scheffe theorem, and Cramer-Rao inequality. Methods: method of moments, maximum likelihood estimation, invariance, least squares estimation (linear regression). Confidence intervals for normal, two-normal, and exponential distributions. |
| Section 10 |
Testing of Hypotheses |
Null/alternative hypotheses, type I & II errors, critical region, significance level, size, power, p-value, most powerful/UMP tests, Neyman-Pearson lemma, and likelihood ratio tests (univariate normal distribution). |
| Section 11 |
Nonparametric Methods |
Tests of randomness (runs test), empirical distribution function, Kolmogorov-Smirnov test (one sample), sign tests (one/two samples), Mann-Whitney test. |
| Section 12 |
Stochastic Processes |
Discrete-time Markov chain: transition probability matrix, higher-order transitions, Chapman-Kolmogorov equation, classification of states/chains, stationary and limiting distributions. Poisson process: properties, interarrival and waiting times. |
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