## MATH-UA.0233.001: Theory of Probability (Spring 2013)

## Goals and topics

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, the Central Limit Theorem and Laws of Large Numbers.

## Logistics

### Classes

WWH 1302, Tuesday and Thursday, 9:30am – 10:45am (75 mins).Attendance of lecture is important: there will often be activities such as worksheets which you will do in pairs or small groups. These will allow you to grapple with concepts and explain them to each other.

For students lying on grade boundaries I will take lecture participation into account.

### Office Hours

Tuesday 3pm-5pm in Room 926 WWH. Office hours are also available by appointment (please make arrangements via e-mail).### Recitations

Room TBA, Friday, 3:30pm – 4:45pm (75 mins).### Text book

A First Course in Probability by Sheldon Ross. The books Introduction to Probability Theory by Hoel, Port and Stone; and Probability and Statistics by Morris DeGroot and Mark Schervish are optional as reference texts. The book Introduction to Probability by C. M. Grinstead and J. L. Snell is available free online.

### Prerequisite

MATH-UA 122 Calculus II and MATH-UA 123 Calculus III with a grade of C or better and/or the equivalent.

### Website

Course materials and other details will be published on NYU classes.

### Tutoring

For free peer tutoring see the University Learning Center website here.

### Homework

Solving a lot of problems is an extremely important part of learning probability.
Homework will be assigned weekly on Thursdays, to be handed in at the start of next Thursday lectures.

In order to get credit, the homework must be submitted to the instructor by the beginning of class on the specified due date. In fairness to the other students in the course, late homework will generally not be accepted. We will, however, drop the lowest homework score in computation of final grades. Please talk to the instructor in cases of emergency.

### Collaboration

I encourage you to study in groups. Get to know others in class. If you have no-one to study with, I can help you find someone.All resources (including the textbooks, the instructor, and computational devices) may be utilized. However you must write up homework in your own words, and understand what you write. Plagiarism (in homework or in exams) is a serious offense (see NYU CAS Academic Policies).

### Quizzes

Quizzes will be given every ... There will be - quizzes. Quizzes will start at the beginning of class. We will drop the lowest quiz.

### Exams

In-class midterm and final; collaboration is forbidden. The midterm exam will take place in class on Thursday, April 4, 2012 at 9:30am-10:45am. The final exam will take place in class on Thursday, May 16, 2012 at 8am-9:50am.

### Grades

Grades will be computed by a weighted average:

Homework | 10% |

Quizzes | 15% |

Midterm | 25% |

Final | 50% |

Total | 100% |

Final scores will be converted to letter grades beginning with the following scale:

93 A |

90 A- |

87 B+ |

83 B |

80 B- |

75 C+ |

65 C |

50 D |

As for a curve, these cutoffs might be adjusted, but only in the downward direction (to make letter grades higher).

### Timeline: (approximate)

Week | Date | Due | Content | ||

1 | Tu | Jan. 29 | Combinatorial Analysis and Story proofs. | ||

Th | Jan. 31 | ||||

2 | Tu | Feb. 5 | Axioms and properties of Probability, Birthday problem. | ||

Th | Feb. 7 | ||||

3 | Tu | Feb. 12 | Conditional Probability, LoTP, Monty Hall, Simpson's paradox | ||

Th | Feb. 14 | Feb 15 is last day to drop courses not receiving a grade of "W". | |||

4 | Tu | Feb. 19 | Gambler's ruin, Random variables and their distributions, | ||

Th | Feb. 21 | Expectations, Indicator random variables. | |||

5 | Tu | Feb. 26 | Poisson Distribution, Discrete vs.~continuous, | ||

Th | Feb. 28 | Uniform Distribution. | |||

6 | Tu | Mar. 5 | Normal Distribution, location, scale and LOTUS | ||

Th | Mar. 7 | ||||

7 | Tu | Mar. 12 | Exponential Distribution, MGFs | ||

Th | Mar. 14 | ||||

8 | Tu | Mar. 19 | No classes. Spring break. | ||

Th | Mar. 21 | No classes. Spring break. | |||

9 | Tu | Mar. 26 | Joint, Conditional and Marginal Distributions, Multinomial | ||

Th | Mar. 28 | and Cauchy distributions, Covariance and correlation. | |||

10 | Tu | Apr. 2 | Midterm Review | ||

Th | Apr. 4 | In class Midterm Exam | |||

11 | Tu | Apr. 9 | Transformations and Convolutions, | ||

Th | Apr. 11 | Beta and Gamma Distributions, Poisson process | |||

12 | Tu | Apr. 16 | Order statistics and conditional distribution | ||

Th | Apr. 18 | ||||

13 | Tu | Apr. 23 | Inequalities, Law of Large numbers and | ||

Th | Apr. 25 | Central Limit Theorem | |||

14 | Tu | Apr. 30 | Chi-square, Student's t and | ||

Th | May 2 | multivariate Normal distribution, Markov chains. | |||

15 | Tu | May 7 | Markov chains and random walks. | ||

Th | May 9 | Final Exam review. | |||

16 | Th | May 16 | Final Exam: 8am-9:50am |