MTH405: (Probability Theory)

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• Sigma algebras, Dynkin's theorem, Probability measure, Independence of events, Borel-Cantelli lemmas.
• Random variables, generated sigma algebra, independence of random variables and Kolmogorov’s zero-one law.
• Distribution functions, density function, examples of standard distributions, jointly distributed random variables, Multivariate distributions (and densities) and properties. Distributions of functions of random vectors and Jacobian formula. Examples of multivariate densities. Conditional and marginal distributions.
• Moments of a random variable, Properties of expectation, variance and covariance, variance-covariance matrix, Cauchy-Schwarz inequality, Markov’s and Chebyshev’s inequality.
• Generating functions, Poisson approximation.
• Convergence of random variables: various notions of convergence and their relations, weak and strong law of large numbers, Kolmogorov's maximal inequality, Kolmogorv three series theorem.
• Weak convergence, Portmanteau theorem, Slutsky’s theorem, Characteristic function and Levy continuity theorem, Central limit theorem, Cramer-Wold device.
• Introduction to Poisson process on [0,$\infty$).

Reference Texts:

1. S. M. Ross: A First Course in Probability.
2. P. Billingsley: Probability and Measure.
3. A. Klenke: ProbabilityTheory, A Comprehensive Course.
4. W. Feller: Introduction to the Theory of Probability and its Applications (Vols. 1 & 2).