Study by subject

Methods of Estimation, Properties of Estimators · Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio Tests · Elementary Bayesian Inference · Analysis of Discrete Data and Chi-Square Test of Goodness of Fit …

Random Variables and Distribution Functions, Expectation and Moments · Characteristic Functions, Probability Inequalities · Modes of Convergence, Laws of Large Numbers, Central Limit Theorem · Sample Space, Discrete Probability, Independent Events, Bayes Theorem

Standard Discrete and Continuous Univariate Distributions · Sampling Distributions, Standard Errors, Asymptotic Distributions · Distribution of Order Statistics and Range

Linear Programming Problem, Simplex Method, Duality · Steady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1) · Elementary Queuing and Inventory Models · Transportation and Assignment Problems …

Simple and Multiple Linear Regression, Logistic Regression · Gauss-Markov Models, Estimability, Best Linear Unbiased Estimators · Analysis of Variance and Covariance, Fixed/Random/Mixed Effects

Multivariate Normal Distribution, Wishart Distribution, Quadratic Forms · Data Reduction: PCA, Discriminant, Cluster Analysis · Partial and Multiple Correlation Coefficients

Connectedness and Orthogonality of Block Designs, BIBD · 2^K Factorial Experiments: Confounding and Construction · Completely Randomized, Randomized Block and Latin-Square Designs · Response Surface Designs

Markov Chains: Classification of States, Stationary Distribution · Poisson and Birth-and-Death Processes · Martingales

Simple Random, Stratified and Systematic Sampling · Ratio and Regression Methods · Probability Proportional to Size Sampling

Descriptive Statistics, Exploratory Data Analysis

Time Series Models

Hazard Function and Failure Rates, Censoring and Life Testing · Series and Parallel Systems

Life Tables, Mortality Rate

Continuity, Uniform Continuity, Differentiability, Mean Value Theorem · Sequences and Series · Metric Spaces, Compactness, Connectedness, Normed Linear Spaces · Monotonic Functions, Bounded Variation, Lebesgue Measure & Integral …

Algebra of Matrices, Rank, Determinant, Linear Equations · Vector Spaces, Subspaces, Linear Dependence, Basis, Dimension · Algebra of Linear Transformations · Inner Product Spaces, Orthonormal Basis …

Finite Differences, Lagrange/Hermite/Spline Interpolation · Numerical Differentiation and Integration · Numerical Solutions of Algebraic Equations, Newton-Raphson, Rate of Convergence · Numerical Solutions of ODEs (Picard, Euler, Runge-Kutta) …

Groups, Subgroups, Sylow Theorems, Cayley's Theorem · Rings, Ideals, UFD, PID, Euclidean Domain · Permutations, Combinations, Pigeon-Hole Principle, Inclusion-Exclusion · Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder Theorem …

Algebra of Complex Numbers, Power Series, Transcendental Functions · Taylor Series, Laurent Series, Calculus of Residues · Liouville's Theorem, Maximum Modulus Principle, Open Mapping Theorem · Conformal Mappings, Mobius Transformations …

General Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's Function · Existence & Uniqueness, Singular Solutions, Systems of First Order ODEs

Lagrange and Charpit Methods, Cauchy Problem for First Order PDEs · Separation of Variables (Laplace, Heat and Wave Equations) · Classification of Second Order PDEs

Separation Axioms, Connectedness and Compactness · Basis, Dense Sets, Subspace and Product Topology

Generalized Coordinates, Lagrange's Equations · Motion of Rigid Bodies, Euler's Dynamical Equations, Small Oscillations · Hamilton's Canonical Equations, Hamilton's Principle

Variation of a Functional, Euler-Lagrange Equation, Conditions for Extrema · Variational Methods for Boundary Value Problems

Fredholm and Volterra Type, Separable Kernels, Resolvent Kernel

Laplace Transform · Fourier Transform

Potential Flow, Streamlines, Doublets

Curves and Surfaces, Curvature

Graphs, Trees, Connectivity

Acceptance Sampling · Control Charts