IIT JAM Mathematical Statistics Syllabus 2025 (Available) – Get MS Syllabus PDF Here

The IIT JAM 2025 syllabus has been officially released online by IIT Delhi, the organizing body for the exam. Candidates can download the syllabus in PDF format to facilitate their preparation. IIT JAM, known as the Joint Admission Test for Masters, is conducted collaboratively by the Indian Institutes of Technology (IITs) and the Indian Institute of Science (IISc).

It serves as a gateway for admission into various postgraduate programs in science and technology disciplines across prestigious institutes in India. The syllabus covers comprehensive topics essential for the exam, aiding candidates in focusing their study efforts effectively.

Students can download the IIT JAM 2025 Syllabus in PDF format. For better preparation, Candidates can also Download other Syllabus in PDF format. The IIT JAM is known as the India Institute of Technology, and the Indian Institute of Science may offer the Joint Admission Test for Masters of Science and Technology.

Click to Download the IIT JAM Mathematical Statistics (MS) Syllabus PDF Here.

Mathematical Statistics 

The Mathematical Statistics (MS) Test Paper is structured with a predominant focus on Statistics, accounting for approximately 70% of its content. The remaining 30% is allocated to Mathematics topics. This distribution highlights the significant emphasis placed on Statistics within the test, ensuring a thorough evaluation of statistical concepts and applications alongside foundational mathematical principles.

Mathematics

Sequences and Series of real numbers: The study of sequences and series in mathematics encompasses fundamental concepts crucial for understanding their convergence and divergence behaviours. Sequences of real numbers involve the analysis of their convergence towards a limit, distinguishing between Cauchy sequences that converge based on closeness criteria and monotonic sequences whose limits are determined by their increasing or decreasing nature.

Infinite series extends this analysis to sums of infinitely many terms, where convergence or divergence depends on the behaviour of terms, often categorized by tests such as comparison, ratio, root, condensation, and integral tests. Absolute convergence ensures convergence regardless of term order, contrasting with conditional convergence seen in alternating series. Power series, meanwhile, explore the convergence within specific radii, essential for applications in functions and calculus.

Differential Calculus of one and two real variables: The study of functions of one and two real variables involves fundamental concepts in calculus and optimization. For functions of one variable, understanding begins with limits, continuity, and differentiability, exploring properties such as Rolle’s theorem and the mean value theorems of Lagrange. Higher-order derivatives extend this analysis, incorporating Leibniz’s rule and applications of Taylor’s theorem, including Maclaurin series and indeterminate forms addressed by L’Hospital’s rule. The study also delves into maxima, minima, and inflexion points, distinguishing between local and global extrema.

Extending to the functions of two variables, the focus shifts to limits, continuity, and differentiability across multiple dimensions. Partial and total differentiation techniques are introduced, complemented by Leibniz’s rule for successive derivatives. Critical points and their characterization using the Hessian matrix identify maxima, minima, and saddle points, crucial for optimization. Constrained optimization further enhances understanding through techniques like the Lagrange multiplier, enabling solutions within specified constraints.

Integral Calculus: Fundamental theorems of integral calculus (single integral). Lebnitz’s rule and its applications. Differentiation under integral sign. Improper integrals. Beta and Gamma integrals:
properties and relationships between them. Double integrals. Change of order of integration. Transformation of variables. Applications of definite integrals. Arc lengths, areas and volumes.

Matrices and Determinants: Vector spaces with real field. Subspaces and sum of subspaces. Span of a set. Linear dependence and independence. Dimension and basis. Algebra of matrices.
Standard matrices (Symmetric and Skew Symmetric matrices, Hermitian and Skew Hermitian matrices, Orthogonal and Unitary matrices, Idempotent and Nilpotent matrices). Definition, properties and applications of determinants. Evaluation of determinants using transformations. Determinant of the product of matrices. Singular and non-singular matrices and their properties. Trace of a matrix. Adjoint and inverse of a matrix and related properties.

The rank of a matrix, row-rank, column-rank, standard theorems on ranks, the rank of the sum and the product of two matrices. Row reduction and echelon forms. Partitioning of matrices and simple properties. Consistent and inconsistent system of linear equations. Properties of solutions of a system of linear equations. Use of determinants in solution to the system of linear equations. Cramer’s rule. Characteristic roots and Characteristic vectors. Properties of characteristic roots and vectors. Cayley Hamilton theorem.

Statistics

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). Geometric probability. Boole’s and Bonferroni’s inequalities. Conditional probability and Multiplication rule. Theorem of total probability and Bayes’ theorem. Pairwise and mutual independence of events.

Univariate Distributions: 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. Mathematical expectations and moments. Mean, Median, Mode, Variance, Standard deviation, Coefficient of variation, Quantiles, Quartiles, Coefficient of Variation, and measures of Skewness and Kurtosis of a probability distribution. Moment generating function (m.g.f.), its properties and uniqueness. Markov and Chebyshev inequalities and their applications.

Standard Univariate Distributions: Degenerate, Bernoulli, Binomial, Negative binomial, Geometric, Poisson, Hypergeometric, Uniform, Exponential, Double exponential, Gamma, Beta (of the first and second type), Normal and Cauchy distributions, their properties, interrelations, and limiting (approximation) cases.

Multivariate Distributions: 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. Distribution of functions of random vectors using transformation of variables and Jacobian method. Mathematical expectation of functions of random vectors. Joint moments, Covariance and Correlation. Joint moment generating function and its properties. Uniqueness of joint m.g.f. and its applications. Conditional moments, conditional expectations and conditional variance. Additive properties of Binomial, Poisson, Negative Binomial, Gamma and Normal Distributions using their m.g.f.

Standard Multivariate Distributions: Multinomial distribution is a generalization of binomial distribution and its properties (moments, correlation, marginal distributions, additive property). Bivariate normal distribution, its marginal and conditional distributions and related properties.

Limit Theorems: Convergence in probability, convergence in distribution and their interrelations. Weak law of large numbers and Central Limit Theorem (i.i.d. case) and their applications.

Sampling Distributions: Definitions of random sample, parameter and statistic. Sampling distribution of a statistic. 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 (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). Central Chisquare 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. Central Student’s 𝒕-distribution: Definition and derivation of p.d.f. of Central Student’s 𝑡-distribution with 𝑛 d.f., Properties and limiting form of central 𝑡-distribution. Snedecor’s Central 𝑭-distribution: Definition and derivation of p.d.f. of snedecor’s Central 𝐹-distribution with (𝑚, 𝑛) d.f.. Properties of Central 𝐹-distribution, distribution of the reciprocal of 𝐹-distribution. Relationship between 𝑡, 𝐹 and 𝜒2 distributions.

Estimation: Unbiasedness. Sufficiency of a statistic. Factorization theorem. Complete statistic. Consistency and relative efficiency of estimators. Uniformly Minimum variance unbiased
estimator (UMVUE). Rao-Blackwell and Lehmann-Scheffe theorems and their applications. Cramer-Rao inequality and UMVUEs. Methods of Estimation: Method of moments, method of maximum likelihood, invariance of maximum likelihood estimators. Least squares estimation and its applications in simple linear regression models. Confidence intervals and confidence coefficient. Confidence intervals for the parameters of univariate normal, two independent normal, and exponential distributions.

Testing of Hypotheses: 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. Most
powerful critical regions and most powerful (MP) tests. Uniformly most powerful (UMP) tests. Neyman Pearson Lemma (without proof) and its applications to the construction of MP and UMP tests for the parameter of single parameter parametric families. Likelihood ratio tests for parameters of univariate normal distribution.

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Sunil Kushwaha

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