MATH 3278 Probability Theory II
The course begins with a discussion of the axioms of probability. We point out that not all subsets of an arbitrary sample can be events and introduce the idea of a sigma field. There is a careful discussion of distribution functions in general (including continuous, absolutely continuous and discrete cases). The rest of the section on distribution theory focuses on the distribution theory of several random variables. Joint density functions, transformations, joint mgfs, order statistics, convolution are discussed. We then define conditional expectation and give its main properties. The section on distribution theory closes with a discussion of multivariate distributions, including the multinomial and multivariate normal. We prove that the sample mean and sample variance in a sample from the normal distribution are independent and obtain the distribution of the sample variance.
The second half of the course focuses on stochastic processes. Markov Chains in discrete time and with discrete state space are discussed. Details are as follows:
Definition of a stochastic process and a Markov Chain; Chapman-Kolmogorov Equations; Classification of states; Ergodic theorem; The Poisson process; Generating functions with Applications to Branching Processes.