Probability Theory, Second Course

Aim

The course gives knowledge of probability theory at an advanced undergraduate level. After completing the course the student will be expected to be able to:

• describe advanced concepts and theorems of probability theory, e.g., different kinds of stochastic convergence and the Cramer-Slutsky theorem, and to prove some of the theorems.
• construct and analyse probabilistic models using advanced tools and methods, e.g., characteristic functions and conditioning.
• describe and prove some basic theorems of statistics.
• understand and assess probabilistic models and analyses occurring in other undergraduate courses, research reports, or the media.
• follow a graduate course in probability theory, and an advanced undergraduate course in statistics.

Content

The transformation theorem. Conditioning. Probability generating function, moment generating function, characteristic function. Order statistics. The multivariate normal distribution, in particular orthogonal transformations and quadratic forms. Convergence concepts, e.g., convergence almost surely, in probability, in r-mean, and in distribution. The Borel-Cantelli lemma and the continuity theorem. The law of large numbers and the central limit theorem. The Cramer-Slutsky theorem.