Details of ID4113 (Autumn 2012)

Level: 4 Type: Theory Credits: 3.0

Course CodeCourse NameInstructor(s)
ID4113 Mathematical Statistics I Asok Kumar Nanda

Syllabus
Mathematical Statistics



Set theory, introduction and revision, limit of a set, union and intersection of countable and uncountable sets, - field and minimal - field, - field generated by a class, Borel set, random variable, basic properties of probability measure, measurable function, functions of a random variable.



pgf, mgf, cgf and their interrelations, uniqueness theorem of mgf, sequence of moments determining a distribution Carlemans criterion, necessary & sufficient conditions for a sequence of moments to uniquely determine a distribution, symmetrization of a random variable. Jacobian of transformation, distribution function of a random variable in two and higher dimensions, derivation of marginal and conditional distributions, discontinuities in a distribution function, decomposition of a distribution function into discrete and continuous parts, multinomial distribution and its properties, distribution of sum, difference, product and quotient of dependent/independent random variables, approximate formulae for expectation and variance of g(X, Y).



Bivariate normal distribution and its properties, independence of sample mean and sample variance, ellipse of concentration, equi-probability contours.



Inequalities basic, Chebyshev, one-sided Chebyshev, Cantelli, Peak, C-S, and Burge inequalities, convex function and Jensen inequality, Holders inequality, Jensens inequality in higher dimension, generalized Holders inequality, Liapounov inequality, -inequality, Minkowskis inequality, Kolmogorovs inequality, convergence of random variables almost sure convergence, convergence in probability, convergence, convergence in Law, moment convergence, Borels 0-1 law, Glivenko Cantelli theorem, empirical distribution function and its properties, mean and variance of sample moments, variance stabilizing transformation, Law of large numbers, central limit theorems- de-Moivre-Laplace, Lindberg-Levy, Liapounov, and Lindeberg-Feller central limit theorems.



Sampling distributions- X2, t, and F distributions, distribution of sample correlation coefficient when population correlation is zero, order statistics from sample, characteristic function and its properties.





References
Suggested Texts/Reference Books:



1. Cramer, H., Mathematical Methods of Statistics, Princeton University Press, 1999.

2. Mood, A., Graybill, F. A. and Boes, D. C., Introduction to the Theory of Statistics (3rd Edition), Mcgraw Hill, 1974.

3. Shao, J., Mathematical Statistics (2nd Edition), Springer, 2003.

4. Rao, C. R., Linear Statistical Inference and Its Applications (2nd Edition), Wiley-Interscience, 2001.

5. Wilks, S. S., Mathematical Statistics, Buck Press, 2008.





Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 1 Not Allowed
2 IP 3 Not Allowed
3 MS 7 Not Allowed
4 RS 1 Not Allowed