# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 107: Probability & Stochastic Processes with Applications to Biology

Approved: 2018-03-01, S. Walcott, M. Goldman
ATTENTION:
Also known as BIS 107.
Units/Lecture:
Lecture—3 hour(s); Laboratory—2 hour(s).
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
"Introduction to Probability Models," 11th edition by Sheldon M. Ross
Search by ISBN on Amazon: 9780124079489
Prerequisites:
(MAT 027A C- or better or BIS 027A C- or better) or (MAT 022A C- or better, (MAT 022AL C- or better or ENG 006 C- or better or ECS 032A C- or better or ECS 036A C- or better or ECH 060 C- or better or EME 005 C- or better)).
Course Description:
Introduction to probability theory and stochastic processes with biological, medical, and bioengineering applications.
Suggested Schedule:
Lecture Section Topics
1 1.1-1.3 Introduction
Sample Spaces and Events
Probabilities Defined on Events
2-3 Counting, Combinations, and Permutations
4-5 1.4-1.6 Conditional Probabilities
Independent Events
Bayes' Formula
6 2.1-2.2 Random Variables
Discrete Random Variables
7 2.3 Continuous Random Variables
8-9.5 2.4 Expectation of a Random Variable
9.5-10 2.5 Jointly Distributed Random Variables
11 Exam
12 3
3.1-3.3
Conditional Probability and Conditional Expectation
Introduction
The Discrete Case
The Continuous Case
13-14 3.4-3.7 Computing Expectations by Conditioning
Computing Probabilities by Conditioning
An Identity for Compound Random Variables
15-16 4
4.1-4.3
Markov Chains
Introduction
Chapman-Kolmogorov Equations
Classification of States
17 4.4 Limiting Probabilities
18 4.5.1 The Gambler's Ruin Problem
19 4.6 Mean Time Spent in Transient States
20 4.7 Branching Processes
21 4.9 Markov Chain Monte Carlo Methods
22 Exam
23 5
5.1-5.2
The Exponential Distribution and the Poisson Process
Introduction
The Exponential Distribution
24 5.3 The Poisson Process
25 10
10.1
Brownian Motion and Stationary Processes
Brownian Motion
26 10.2 Hitting Times, Maximum Variable, and the Gambler's Ruin Problem
27 10.3 Variations on Brownian Motion
28 Buffer lecture

### Laboratory Content

Lab Biology Example
1 Simple sequence alignment; combinatories
2 Mark and recapture
2 Screening for Down's syndrome in the first trimester
3 Sanger sequencing
4 Multivariate distributions
4 Mouse behavior
5 RNA-Seq
6 Breast cancer
7 SARS outbreak
8 Hidden Markov models
9 Population dynamics