# 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 |