# Syllabus: Sampling Distribution and Order Statistics

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Group A: Sampling Distribution
Functions of random variables and random vectors: Method of distribution functions, method of transformation, Method of moment generating functions, Probability integral transform.
Distribution of sum, difference, product and quotient or random variables, functions of random vectors of continuous and discrete type, Central limit theorem with applications, Inversion theorem.

Sampling Distributions: Definition, examples from discrete and continuous populations, difference from probability distribution.

Different method of finding sampling distribution: Analytic method, inductive method, geometric method, method of using characteristic function etc. Distribution of sample mean and variance and their independence for normal population, chi- square (χ2), F and t distributions (central and non-central cases), their uses in statistics. Sampling distribution of correlation and regression coefficients, derivation of frequency χ2.
Standard errors of statistics and their large sample approximations. Transformation of variables including square root, log, sin-inverse etc.

Group B: Order Statistics
Definition and distribution of functions of order statistics for bot discrete and continuous cases, asymptotic distributions, sample cumulative distribution functions, distribution of a single order statistics, joint distribution of order statistics, distribution of range and some other statistics.