## Masters Theses & Specialist Projects

Spring 2018

#### Advisor(s) - Committee Chair

David Neal (Director), Melanie Autin, and Ngoc Nguyen

#### Degree Program

Department of Mathematics

#### Degree Type

Master of Science

#### Abstract

The Bernoulli distribution is a basic, well-studied distribution in probability. In this thesis, we will consider repeated Bernoulli trials in order to study runs of identical outcomes. More formally, for t ∈ N, we let Xt ∼ Bernoulli(p), where p is the probability of success, q = 1 − p is the probability of failure, and all Xt are independent. Then Xt gives the outcome of the tth trial, which is 1 for success or 0 for failure. For n, m ∈ N, we define Tn to be the number of trials needed to first observe n consecutive successes (where the nth success occurs on trial XTn ). Likewise, we define Tn,m to be the number of trials needed to first observe either n consecutive successes or m consecutive failures.

We shall primarily focus our attention on calculating E[Tn] and E[Tn,m]. Starting with the simple cases of E[T2] and E[T2,2], we will use a variety of techniques, such as counting arguments and Markov chains, in order to derive the expectations. When possible, we shall also provide closed-form expressions for the probability mass function, cumulative distribution function, variance, and other values of interest. Eventually we will work our way to general formulas for E[Tn] and E[Tn,m]. We will also derive formulas for conditional averages, and discuss how famous results from probability such as Wald’s Identity apply to our problem. Numerical examples will also be given in order to supplement the discussion and clarify the results.

#### Disciplines

Applied Mathematics | Discrete Mathematics and Combinatorics | Mathematics

COinS