State University College at Buffalo

Mathematics Department

Request for Course

I. Number and Title of course

MATH 681 - Intermediate Probability
II. Reasons for the addition to the present curricula.
A. To provide a graduate level course in Mathematical statistics.

B. To provide for these students who have had a one-semester introduction to Mathematical probability at the undergraduate level, an opportunity to extend, deepen and broaden this knowledge at the graduate level, and to acquaint themselves with the richness of application.

III. Major Objectives of the Course
A. To review, but more importantly deepen, the student's elementary acquaintance by an examination of the more involved aspects of probability theory.

B. To introduce and develop more sophisticated mathematical models for the study of random phenomena than are encountered in the undergraduate introduction to Mathematical Probability.

C. To produce computational methods and formulae appropriate to the evaluation of desired probabilities in a specific application of a general stochastic model.

IV. Topical Outline
Topics which may be selected by the instructor as appropriate for an integrated graduate course and which are more advanced than those encountered in one-semester introduction to Probability theory include

A. Combinatorial Analysis

1. Occupancy Problems.

2. Problem of Runs

3. Matching Problems

4. Simple Random Walk Problems

5. Fluctuation Theory

B. Laws of large Numbers
1. Infinite Sequences of Bernoulli Trials

2. Borel-Cantelli Lemmas

3. Weak and Strong Laws of Large Numbers

4. Central Limit Theorem

5. Laws of the Iterated Logarithm

C. Stochastic Processes
1. Markov Chain

2. Branching and Queuing.

3. Birth and Death Processes

4. Renewal Theory

5. Gaussian and Wiener Processes