JAM, the Joint Admission Test for Masters, is the entrance examination for M.Sc. and related programmes at the IITs and other institutes; Mathematical Statistics (MS) is its statistics paper. The paper has twelve sections: three of mathematics (sequences and series, calculus, matrices) and nine of statistics. This page maps the JAM 2026 MS syllabus document. Each line is reproduced as the document words it, then pointed at the course unit here that teaches it, with the depth stated.
No exam pattern is stated here. The syllabus document gives topics only. Marks, question types, duration, negative marking and eligibility are in the official JAM information brochure for the year you sit; read that for the rules, and this page for the statistics.
deep a full unit page here, with the derivation worked out and problems solved step by step — enough to answer a descriptive paper, not just recognise the term. brief covered, but at exam-summary level: the definition, the formula and a worked example, without the derivation. not here nothing on this site teaches it. It is named rather than quietly skipped, because a map that hides its holes only tells you about them in the hall.
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| Syllabus line, as prescribed | Where it is taught here | Depth |
|---|---|---|
| Sequences of real numbers, their convergence, and limits. Cauchy sequences and their convergence. Monotonic sequences and their limits. Limits of standard sequences. | Mathematical Analysis (Unit 4) — Sequences and Series of Real Numbers: the Tests of Convergence UGC NET Unit II — Sequences of Real Numbers | deep |
| Limit superior and limit inferior of sequences. | Mathematical Analysis (Unit 4) — Sequences and Series of Real Numbers: the Tests of Convergence | deep |
| Infinite series and its convergence, and divergence. Convergence of series with non-negative terms. Tests for convergence and divergence of a series. Comparison test, limit comparison test, D’Alembert’s ratio test, Cauchy’s nth root test, Cauchy’s condensation test, and integral test. | Mathematical Analysis (Unit 4) — Sequences and Series of Real Numbers: the Tests of Convergence UGC NET Unit II — Series of Real Numbers | deep |
| Absolute convergence of series. Leibnitz’s test for the convergence of alternating series. Conditional convergence. | Mathematical Analysis (Unit 4) — Sequences and Series of Real Numbers: the Tests of Convergence | deep |
| Convergence of power series and radius of convergence. | Mathematical Analysis (Unit 4) — Sequences and Series of Real Numbers: the Tests of Convergence UGC NET Unit II — Power Series and Radius of Convergence | deep |
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| Syllabus line, as prescribed | Where it is taught here | Depth |
|---|---|---|
| Differential Calculus of one variable: Limits of functions of one real variable. Continuity and differentiability of functions of one real variable. Properties of continuous and differentiable functions of one real variable. | Mathematical Analysis (Unit 1) — Limits and Continuity UGC NET Unit II — Functions of a Real Variable | deep |
| Rolle’s theorem and Lagrange’s mean value theorems. | Mathematical Analysis (Unit 1) — Differentiation UGC NET Unit II — Differentiability and Mean Value Theorems | deep |
| Higher order derivatives, Lebnitz’s rule and its applications. Taylor’s theorem with Lagrange’s and Cauchy’s form of remainders. Taylor’s and Maclaurin’s series of standard functions. Indeterminate forms and L’ Hospital’s rule. Maxima and minima of functions of one real variable, critical points, local maxima and minima, global maxima and minima, and point of inflection. | Mathematical Analysis (Unit 1) — Calculus of One Variable: Successive Derivatives, Taylor Series and Extrema | deep |
| Differential calculus of two variables: Limits of functions of two real variables. Continuity and differentiability of functions of two real variables. Properties of continuous and differentiable functions of two real variables. Partial differentiation and total differentiation. | Mathematical Analysis (Unit 1) — Calculus of Two Variables: Total Differential, Hessian and Lagrange Multipliers UGC NET Unit II — Functions of Two Real Variables | deep |
| Lebnitz’s rule for successive differentiation. | Mathematical Analysis (Unit 1) — Calculus of One Variable: Successive Derivatives, Taylor Series and Extrema | deep |
| Maxima and minima of functions of two real variables. Critical points, Hessian matrix, and saddle points. Constrained optimization techniques (with Lagrange multiplier). | Mathematical Analysis (Unit 1) — Calculus of Two Variables: Total Differential, Hessian and Lagrange Multipliers | deep |
| Integral Calculus: Fundamental theorems of integral calculus (single integral). | UGC NET Unit II — Riemann Integration and Improper Integrals | brief |
| Lebnitz’s rule and its applications. Differentiation under integral sign. | Mathematical Analysis (Unit 3) — Differentiation Under the Integral Sign | deep |
| Improper integrals. | UGC NET Unit II — Riemann Integration and Improper Integrals | brief |
| Beta and Gamma integrals: properties and relationship between them. | Continuous Distributions (Unit 3) — Preliminaries: Gamma Function Mathematical Analysis (Unit 3) — Double Integrals, Change of Variables, Arc Length, Area and Volume | deep |
| Double integrals. Change of order of integration. Transformation of variables. | Mathematical Analysis (Unit 3) — Double Integrals, Change of Variables, Arc Length, Area and Volume Mathematical Analysis (Unit 3) — Interchanging the Order of Integration | deep |
| Applications of definite integrals. Arc lengths, areas and volumes. | Mathematical Analysis (Unit 3) — Double Integrals, Change of Variables, Arc Length, Area and Volume | deep |
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| Syllabus line, as prescribed | Where it is taught here | Depth |
|---|---|---|
| R^n and C^n as vector spaces over real field. Span of a set. Linear dependence and independence. Dimension and basis. Null space. | UGC NET Unit II — Vector Spaces, Span, Linear Independence, Basis UGC NET Unit II — Rank, Nullity, Row Reduced Echelon Form | brief |
| Algebra of matrices. | UGC NET Unit II — Trace, Determinant, Inverse | brief |
| Standard matrices (Symmetric and Skew Symmetric matrices, Hermitian and Skew Hermitian matrices, Orthogonal and Unitary matrices, Idempotent and Nilpotent matrices). | Linear Algebra & Linear Models (Unit 1) — Special Matrices, Determinants and Rank UGC NET Unit II — Symmetric, Skew-symmetric, Orthogonal Matrices | deep |
| Definition, properties and applications of determinants. Evaluation of determinants using transformations. Determinant of product of matrices. Singular and non-singular matrices, and their properties. | Linear Algebra & Linear Models (Unit 1) — Special Matrices, Determinants and Rank UGC NET Unit II — Trace, Determinant, Inverse | deep |
| Trace of a matrix. Adjoint and inverse of a matrix, and related properties. | UGC NET Unit II — Trace, Determinant, Inverse | brief |
| Rank and nullity of a matrix, row-rank, column-rank, standard theorems on ranks, rank of the sum and the product of two matrices. | Linear Algebra & Linear Models (Unit 1) — Special Matrices, Determinants and Rank UGC NET Unit II — Rank, Nullity, Row Reduced Echelon Form | deep |
| Row reduction and echelon forms. | UGC NET Unit II — Rank, Nullity, Row Reduced Echelon Form | brief |
| Consistent and inconsistent systems of linear equations. Properties of solutions of system of linear equations. Use of determinants in solving the system of linear equations. Cramer’s rule. | Linear Algebra & Linear Models (Unit 1) — Solving Systems of Linear Equations Linear Algebra & Linear Models (Unit 1) — Special Matrices, Determinants and Rank | deep |
| Characteristic roots and Characteristic vectors. Properties of characteristic roots and vectors. Cayley-Hamilton theorem. | Linear Algebra & Linear Models (Unit 2) — Characteristic Roots and Vectors Linear Algebra & Linear Models (Unit 2) — The Cayley–Hamilton Theorem | deep |
| Quadratic forms, positive definite, positive semi-definite, negative definite, and negative semi-definite matrices, and their simple properties. | Linear Algebra & Linear Models (Unit 3) — Real Quadratic Forms | deep |
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| Syllabus line, as prescribed | Where it is taught here | Depth |
|---|---|---|
| Descriptive Statistics: Concepts of sample and population. Different types of data. Tabular and graphical representation of data. | Descriptive Statistics (Unit 1) Descriptive Statistics (Unit 2) | deep |
| Measures of central tendency (arithmetic mean, geometric mean, harmonic mean, median, mode). | Descriptive Statistics (Unit 3) | deep |
| Measures of dispersion (range, inter quartile range, mean deviation about a point, standard deviation, variance, coefficient of variation). | Descriptive Statistics (Unit 4) | deep |
| Moments, central moments, skewness and kurtosis. | Descriptive Statistics (Unit 5) | deep |
| Bivariate data: Scatter diagram, covariance, simple, partial and multiple correlations (3 variables only), Spearman’s rank correlation. | Statistical Methods (Unit 2) Statistical Methods (Unit 3) | deep |
| Probability: Random Experiments. Sample Space and Algebra of Events (Event space). Relative frequency and Axiomatic definitions of probability. Properties of probability function. Addition theorem of probability function (inclusion-exclusion principle). | Theory of Probability (Unit 1) — Basic Concepts Theory of Probability (Unit 1) — Addition Theorem of Probability | deep |
| Geometric probability. | Theory of Probability (Unit 1) — Geometric Probability | deep |
| Boole’s and Bonferroni’s inequalities. | Theory of Probability (Unit 1) — Boole's Inequality | deep |
| Conditional probability and Multiplication rule. Theorem of total probability and Bayes’ theorem. Pairwise and mutual independence of events. | Theory of Probability (Unit 1) — Conditional Probability Theory of Probability (Unit 1) — Bayes' Theorem | deep |
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| Syllabus line, as prescribed | Where it is taught here | Depth |
|---|---|---|
| Definition of random variables. Cumulative distribution function (c.d.f.) of a random variable. Discrete and Continuous random variables. Probability mass function (p.m.f.) and Probability density function (p.d.f.) of a random variable. Distribution (c.d.f., p.m.f., p.d.f.) of a function of a random variable using transformation of variable and Jacobian method. | Theory of Probability (Unit 2) | deep |
| Mathematical expectation and moments. Mean, Median, Mode, Variance, Standard deviation, Coefficient of variation, Quantiles, Quartiles, and measures of Skewness and Kurtosis of a probability distribution. | Theory of Probability (Unit 4) Theory of Probability (Unit 2) — Moments from PMF / PDF | deep |
| Moment generating function (m.g.f.), its properties and uniqueness. | Theory of Probability (Unit 5) — Moment Generating Function (MGF) | deep |
| Markov and Chebyshev inequalities, and their applications. | Theory of Probability (Unit 4) — Chebyshev's Inequality | deep |
| Degenerate, Bernoulli, Binomial, Negative binomial, Geometric, Poisson, Hypergeometric, Uniform, Exponential, Double exponential, Gamma, Beta (of first and second type), Normal and Cauchy distributions, their properties, interrelations, and limiting (approximation) cases. | Theoretical Discrete Distributions Theoretical Continuous Distributions Distribution Theory (Unit 1) | deep |
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| Definition of random vectors. Joint and marginal c.d.f.s of a random vector. Discrete and continuous type random vectors. Joint and marginal p.m.f., joint and marginal p.d.f.. Conditional c.d.f., conditional p.m.f. and conditional p.d.f. Independence of random variables. | Theory of Probability (Unit 3) | deep |
| Distribution of functions of random vectors using transformation of variables and Jacobian method. | Theory of Probability (Unit 3) — Transformation of Two Random Variables (Jacobian Method) Distribution Theory (Unit 2) — Functions of Random Variables: the Jacobian Rule | deep |
| Mathematical expectation of functions of random vectors. Joint moments, Covariance and Correlation. | Theory of Probability (Unit 4) — Covariance via Expectation | deep |
| Joint moment generating function and its properties. Uniqueness of joint m.g.f. and its applications. | Theory of Probability (Unit 5) — Joint Moment Generating Function | deep |
| Conditional moments, conditional expectations and conditional variance. | Theory of Probability (Unit 4) Probability Theory (Unit 2) — Conditional Expectation and Conditional Variance | deep |
| Additive properties of Binomial, Poisson, Negative Binomial, Gamma and Normal Distributions using their m.g.f. | Discrete Distributions (Unit 2) — Additive Property Continuous Distributions (Unit 3) — Gamma Distribution Continuous Distributions (Unit 4) — Additive Property | deep |
| Multinomial distribution as a generalization of binomial distribution and its properties (moments, correlation, marginal distributions, additive property). | Discrete Distributions (Unit 1) — Multinomial Distribution | deep |
| Bivariate normal distribution, its marginal and conditional distributions and related properties. | Continuous Distributions (Unit 4) — Bivariate Normal Distribution | deep |
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| Convergence in probability, convergence in mean square, almost sure convergence, convergence in distribution, and their inter-relations. | Theory of Probability (Unit 5) — Modes of Convergence Probability Theory (Unit 3) — The Implications Between the Modes | deep |
| Weak law of large numbers, Strong law of large numbers, and Central Limit Theorem (i.i.d. and finite variance case). | Theory of Probability (Unit 5) — Weak Law of Large Numbers (WLLN) Theory of Probability (Unit 5) — Central Limit Theorem (CLT) | deep |
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| Definitions of random sample, parameter and statistic. Sampling distribution of a statistic. | Continuous Distributions (Unit 5) — Population, Sample, Parameter, Statistic | deep |
| Order Statistics: Definition and distribution of the rth order statistic (d.f. and p.d.f. for i.i.d. case for continuous distributions). | Distribution Theory (Unit 4) — Order Statistics | deep |
| Distribution (c.d.f., p.m.f., p.d.f.) of smallest and largest order statistics (i.i.d. case for discrete as well as continuous distributions). | Distribution Theory (Unit 4) — The Smallest and Largest Order Statistics, Including the Discrete Case | deep |
| Central Chi-square distribution: Definition and derivation of p.d.f. of central χ² distribution with n degrees of freedom (d.f.) using m.g.f. Properties of central χ² distribution, additive property and limiting form of central χ² distribution. | Continuous Distributions (Unit 5) — Chi-square (χ²) Distribution | deep |
| Central t-distribution: Definition and derivation of p.d.f. of Central t-distribution with n d.f., Properties and limiting form of central t-distribution. | Continuous Distributions (Unit 5) — Student's t-Distribution | deep |
| Central F-distribution: Definition and derivation of p.d.f. of Central F-distribution with (m,n) d.f. Properties of Central F-distribution, distribution of the reciprocal of F-distribution. | Continuous Distributions (Unit 5) — F-Distribution (Snedecor's F) | deep |
| Relationship between t, F and χ² distributions. | Continuous Distributions (Unit 5) — Relationships among Sampling Distributions | deep |
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| Null and alternative hypotheses (simple and composite), Type-I and Type-II errors. Critical region. Level of significance, size and power of a test, p-value. | Inferential Statistics (Unit 2) — Statistical Hypothesis Inferential Statistics (Unit 2) — Two Types of Errors Inferential Statistics (Unit 2) — p-value | deep |
| Most powerful critical regions and most powerful (MP) tests. Uniformly most powerful (UMP) tests. Neyman-Pearson Lemma (without proof) and its applications to construction of MP and UMP tests for parameter of one-parameter parametric families. | Inferential Statistics (Unit 2) — Neyman–Pearson Lemma Testing of Hypotheses (Unit 2) — Uniformly Most Powerful Tests | deep |
| Likelihood ratio tests for parameters of univariate normal distribution. | Inferential Statistics (Unit 2) — Generalized Likelihood Ratio Test (LRT) Testing of Hypotheses (Unit 3) | deep |
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| Tests of randomness based on total number of runs. | Inferential Statistics (Unit 5) — One-Sample Runs Test (Test for Randomness) | deep |
| Empirical distribution function. | Probability Theory (Unit 3) — The Glivenko–Cantelli Lemma | deep |
| Kolmogorov-Smirnov one sample test. | Inferential Statistics (Unit 5) — Kolmogorov–Smirnov (K–S) Test | deep |
| One and two sample sign tests. | Inferential Statistics (Unit 5) — Sign Test | deep |
| Mann-Whitney test. | Inferential Statistics (Unit 5) — Wilcoxon–Mann–Whitney U Test | deep |
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| Syllabus line, as prescribed | Where it is taught here | Depth |
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| Discrete time Markov chain: transition probability matrix, higher order transition probabilities, Markov chain as a graph, Chapman-Kolmogorov equation, classification of states and chains, stability of Markov chain (stationary and limiting distributions). | UGC NET Unit IX — Markov Chains: Definitions UGC NET Unit IX — Classification of States UGC NET Unit IX — Stationary Distribution | deep |
| Poisson process and its properties, interarrival and waiting times. | UGC NET Unit IX — Poisson Process UGC NET Unit IX — Inter-arrival and Waiting Time Distributions | deep |
No gaps in the JAM MS syllabus. Every line above points at a page on this site. Lines marked brief are covered at exam-summary level in UGC NET Unit II; the course units linked beside them carry the full treatment of the neighbouring lines.
The courses this exam’s map sends you to, in learning order — 12 of them. Each is listed because the map links into it, not because it was judged relevant, so a course missing here is one no line of this syllabus points at.