NYU Stern School of Business

Undergraduate College


Fall 2011

Instructor Details

Melnick, Edward


212 998 0444


KMC 8-56


Course Meetings

R, 6:00pm to 9:00pm

Tisch T-201

Final Exam:

Schedule exceptions
    Class will not meet on:
    Class will meet on:


Course Description and Learning Goals


Courses offered in Statistics can be broadly categorized as methodological with emphasis on applications and theory. The course Introduction to the Theory of Probability is a THEORY course. It is designed to present the major ideas of probability distribution theory assuming a mathematical background consisting of two semesters of calculus and one semester of linear algebra. The course should serve as background preparation for many courses including applied statistics, stochastic processes, mathematical finance, and actuarial science. In particular, the course will cover concepts needed for 50% of the first actuarial exam: Mathematical Foundations of Actuarial Science (the remaining 50% of the exam is Calculus). It is the prerequisite course for Stochastic Processes I, Statistics and Regression Analysis, and is a required course in the Ph.D. Core. The course emphasis will be the formulation of probability models in order to describe underlying generating processes. All concepts will be illustrated with examples and presented to the students in the form of problems to be solved. Required material will be presented in class and students will be urged to participate in the development of these concepts.


Course Pre-Requisites

V63.0121 Calculus1 and V63.0122 Calculus II


Course Outline


Reading by topic for each text, and homework problems assigned from (HC).



 1. General Concepts of Probability

(HC) Chapter 1, Sections 1-4

(HT) Chapters 1, 2

(R) Chapter 1


Page 20: 11-16


2. Random Variables, Probability Functions

And Distribution Functions


(HC) Chapter 1, Sections 5-7

(HT) Chapter 3, Sections 2-3

(HT) Chapter 4, Pgs. 180-184

(R) Chapter 2, Pgs. 33-35 and 46-48


Page 40: 1, 2, 4, 5, 8

Page 44: 1, 2, 3

Page 51: 3, 7, 10, 20, 21, 22, 23


3. Mathematical Expectation

(HC) Chapter 1, Sections 8-10

(HT) Chapter 3, Sections 2-3

(HT) Chapter 4, Pgs. 184-188

(R) Chapter 4, Sections 1-2

Page 57: 3, 4, 8

Page64: 1-8,17,18

Page72: 2, 3



 4. Multivariate Distributions

(HC) Chapter 2

(HT) Chapter 11, Sections 1-3

(R) Chapter 3, Sections 1-6

(R) Chapter 4, Sections 3-4


Page 83: 6, 7, 8

Page 92: 1, 2. 3, 4

Page 99: 1, 2, 3, 4, 10

Page 107: 1, 3, 7

Page 114: 2, 4, 8, 9

Page 122: 3, 8


5. Special Distributions

(HC) Chapter 3, Sections 1-2 (Discrete)

(HC) Chapter 3, Section 3 and 6 (Normal Related)

(HC) Chapter 3, Section 4 (Normal not 3, 4.1)

(HC) Chapter 3, Section 5 (Multivariate Normal not 3, 5.1)

1. Let X1, X2, and X3 be random variables with equal variances but with correlation coefficients r12 = 0.3, r13 = 0.5 and r23 = 0.2.  Find the correlation coefficient of the linear functions Y = X1 + X2 and Z = X2 + X3.

2. Find the variance of the sum of 10 random variables if each has variance 5 and if each pair has correclation coefficient 0.5

(HT) Chapter 3, Sections 4-7

(HT) Chapter 4, Sections 2-4

(HT) Chapter 11, Sections 4-5

(R) Chapter 2, Sections 1-2


 Page 140: 1-6, 18, 19

Page 147: 1-5, 8

Page 157: 1-5, 18-21

Page 168: 2-6, 9-14

Page 179: 2, 5, 10

Extra Problems


6. Distribution of Functions of Random Variables

(HC) Chapter 4

(HT) Chapter 3, Section 8

(HT) Chapter 4, Section 5

(HT) Chapter 5, Sections 1-3

(R) Chapter 2, Sections 3-4

(R) Chapter 3, Sections 7-8

(R) Chapter 4, Sections 5-6

(R) Chapter 6

(R) Chapter 11, Section 6



Page 188: 8 - 11

Page 201: 4, 5, 9, 16

Let X1 and X2  be independent random variables.  Let X1 and Y = X1 + X2  have chi-square distributions with r1 and r degrees of freedom, respectively.  Here r1 < r.  Show that X2 has a chi-square distribution with rr1 degrees of freedom. 

Hint:  Write M (t) = E(e1(x1+x2)  and make use of the independence of X.  



7. Limit Theorems

(HC) Chapter 5

(HT) Chapter 5, Sections 4-6

(R) Chapter 5


Page 247: 5, 6

Page 225: 1 -3, 5, 7, 9


8. Point Estimation

(HC) Chapter 6, Section 1

(HT) Chapter 6, Section 1

(R) Chapter 8, Section 1, up to Pg. 258


Page 318: 1, 3, 6

Homework Set 1

1. How many two-digit numbers can be formed with the integers 1, 2, 3, 4, 5, if duplication of the integers is not allowed? If duplication is allowed?

2. How many three-digit numbers can be formed from 0, 1, 2, 3, 4, if duplication is not allowed? How many of these are even?

3. There are six roads from A to B and three roads from B to C. In how any ways can one go from A to C via B?

4. How many different sums of money can be formed with at most one of the six kinds of coins minted by the United States Treasury (penny, nickel, dime, quarter, half dollar, and dollar)?

5. In a baseball league of eight teams, how many games will be necessary if each team is to play every other team twice at home?

6. How many signals can a ship show with five different flags if there are five significant positions on the flagpole?

7. Six dice are tossed. What is the probability that every possible number will appear?

8. Seven dice are tossed. What is the probability that every number appears

9. An urn contains four white and five black balls; a second urn contains five white and four black ones. One ball is transferred from the first to the second urn; then a ball is drawn from the second urn. What is the probability that it is white?

10. In the above problem suppose that two balls, instead of one, are transferred from the first to the second urn. Find the probability that a ball then drawn from the second urn will be white.

11. The game of craps is played with two dice as follows: In a particular game one person throws the dice. He wins on the first throw if he gets 7 or 11 points; he loses on the first throw if he gets 2, 3, or 12 points. If he gets 4, 5, 6, 8 9, or 10 points on the first throw, he continues to throw the dice repeatedly until he produces either a 7 or the number first thrown; in the latter case he wins, in the former he loses. What is his probability of winning?


Required Course Materials


You may select one of the three texts for the course. The homework problems will be from Hogg and Craig 6th edition.


Assessment Components


Problem assignments will be given at the end of each session and will be submitted the following week for review. All assignments must be completed. Each week two problems from the homework will be graded. The final grade is computed as:



Classroom Participation


Graded Homework Problems


Final Examination




At NYU Stern we seek to teach challenging courses that allow students to demonstrate their mastery of the subject matter.  In general, students in undergraduate core courses can expect a grading distribution where: 

Note that while the School uses these ranges  as a guide, the actual distribution for this course and your own grade will depend upon how well  you actually perform in this course.


Professional Responsibilities For This Course




In-class contribution is a significant part of your grade and an important part of our shared learning experience. Your active participation helps me to evaluate your overall performance.
You can excel in this area if you come to class on time and contribute to the course by:




Classroom Norms


Stern Policies

General Behavior
The School expects that students will conduct themselves with respect and professionalism toward faculty, students, and others present in class and will follow the rules laid down by the instructor for classroom behavior. 

Collaboration on Graded Assignments
Students may work together on assignment but each person is expected to submit his/her own work.


Academic Integrity

Integrity is critical to the learning process and to all that we do here at NYU Stern. As members of our community, all students agree to abide by the NYU Stern Student Code of Conduct, which includes a commitment to:

The entire Stern Student Code of Conduct applies to all students enrolled in Stern courses and can be found here:

Undergraduate College: http://www.stern.nyu.edu/uc/codeofconduct
Graduate Programs: http://w4.stern.nyu.edu/studentactivities/involved.cfm?doc_id=102505

To help ensure the integrity of our learning community, prose assignments you submit to Blackboard will be submitted to Turnitin.  Turnitin will compare your submission to a database of prior submissions to Turnitin, current and archived Web pages, periodicals, journals, and publications.  Additionally, your document will become part of the Turnitin database.


Recording of Classes

Your class may be recorded for educational purposes


Students with Disabilities

If you have a qualified disability and will require academic accommodation of any kind during this course, you must notify me at the beginning of the course and provide a letter from the Moses Center for Students with Disabilities (CSD, 998-4980, www.nyu.edu/csd) verifying your registration and outlining the accommodations they recommend.  If you will need to take an exam at the CSD, you must submit a completed Exam Accommodations Form to them at least one week prior to the scheduled exam time to be guaranteed accommodation.


Group Projects

Guidelines for Group Projects

Business activities involve group effort. Consequently, learning how to work effectively in a group is a critical part of your business education.

Every member is expected to carry an equal share of the group’s workload. As such, it is in your interest to be involved in all aspects of the project. Even if you divide the work rather than work on each piece together, you are still responsible for each part. The group project will be graded as a whole:   its different components will not be graded separately. Your exams may contain questions that are based on aspects of your group projects.

It is recommended that each group establish ground rules early in the process to facilitate your joint work including a problem-solving process for handling conflicts. In the infrequent case where you believe that a group member is not carrying out his or her fair share of work, you are urged not to permit problems to develop to a point where they become serious. If you cannot resolve conflicts internally after your best efforts, they should be brought to my attention and I will work with you to find a resolution.

You will be asked to complete a peer evaluation form to evaluate the contribution of each of your group members (including your own contribution) at the conclusion of each project. If there is consensus that a group member did not contribute a fair share of work to the project, I will consider this feedback during grading.



The process of assigning grades is intended to be one of unbiased evaluation. Students are encouraged to respect the integrity and authority of the professor’s grading system and are discouraged from pursuing arbitrary challenges to it.

If you believe an inadvertent error has been made in the grading of an individual assignment or in assessing an overall course grade, a request to have the grade re-evaluated may be submitted. You must submit such requests in writing to me within 7 days of receiving the grade, including a brief written statement of why you believe that an error in grading has been made.


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