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
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 |