MAT 330: Probability.

- Practice Exam for test Nov 12
- Test - redo for hw to show you know material

- The Final Exam will be given during Finals Week on Tuesday, Dec 17 from 3:45-5:45
- Do the above for practice . I also have an alternate midterm you might look at.
- More Review problems for final. These are topics covered after the midterm.
- Please note that I have cleaned up notes for the proofs I tried to present on the tablet on Dec 12 when I had technical difficulties. I have also given some simpler proofs of the main theorems. These are at the bottom of this page.
- Graduate students will be expected to totally understand the proof of Var(X1+X2+...+Xn)= Var(X1)+Var(X2)+..Var(Xn) when X1,..Xn are pairwise independent. Look at notes at bottom of the page. I have given a basic outline of a simpler proof under the Dec 12 class notes. You have to understand all the details and make an appointment to give an oral presentation to me. The class notes present a more general but intuitive theorem that when you take functions of independent variables you still have independence .

I will give you a selection problems for the material we have covered since the tests since the midterm on Satyrdat

4 hours, 4 credits. Basic probability theory. Combinatorial problems, distributions, expectation, law of large numbers and central limit theorem, Bernoulli processes, and Markov chains. Other topics from probability and statistics. PREREQ: MAT 176.

MAT 681: Probability.

4 hours, 4 credits. Probability models, combinatorial problems, random variables, expectation and variance, binomial, normal and Poisson variables, law of large numbers, central-limit theorem, markov chains, and selected additional topics. PREREQ: Two semesters of calculus.

Location:Gi 225 TTH 4:00-5:40

Instructor:Robert Schneider

Contact Info:

**email:**robert.schneider@lehman.cuny.edu**web page**: comet.lehman.cuny.edu/schneider/**office hours**: T,TH 2-3 Gillet 211+ by appointment - administrative meetings; often 3-3:50PM TTH

Grading Policy:

- Homework test (half/three quarter hour) on one Tuesday each month 20%
- Midterm - 30 %
- Final: 40%-- May 22 11:00 in Gi 205 -- Combination of two tests- 2 hours
- Homework- 10%
- extra credit will be given for various projects and exceptional class comments

Course Objectives:

- understand fundamental theorems and assumptions underlying probability
- prove some of the fundamental theorems
- apply appropriate theorems from probability

Materials, Resources and Accommodating Disabilities:

textbook:A First Course in Probability; Sheldon Ross; 8th or 9th edition; Pearson

- Classic Texts

- An Introduction to Probability Theory and Its Applications, William Feller; Vol. 1, 3rd Edition; Wiley-- one of great texts; very complete and compact; good for graduate students to look at
- Probability: A Survey of the Mathematical Theory (Wiley Series in Probability and Statistics);John W. Lamperti; 2 edition (August 23, 1996)
- Schaum's Outline Of Probability And Statistics, 4th Edition;Mcgraw Hill -- lots of problems concise; cheap
Accommodating Disabilities:Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.## Course Calendar:

We will try to cover the material in Chapters 1-5 of the book at a speed appropriate for maximal understanding and retention. Continuous distributions will be introduced early and I may deviate or skip around in book to cover these. I may introduce some topics from chap 6 and 7 if time permits. Graduate students will be required to read these chapters and contact me for help. I will not accept homework late.

- 8/28-- Chapter 1

- Problems (both editions different pages but same problems at end of chapter) -due Tue Sep 3.

- Problems section : 1,2,4,7
- Theoretical Section:2
- Grad students add:Theoretical 3,4,8,13
- 9/3 Ch1 start of Ch 2

- Problems: Ch 1: 8,9,10,24,26; Ch 2: 1,2,3,4 (hand in second set on Thur 9/12)
- 9/10 Ch2

- Problems 8,9,10,13,15,16,19,29 ; Theoretical exercises 1-8 (due 9/24)
- Grad Students should pick a topic below and make appointment to discuss concepts with me. I will give more topics and you will only need 3 discussions.
- 9/30 Ch 3

- Due Oct 8: 3.1,3.2,3.4,3.5 + Explain how we get probability of a length between 0 and 1 when we are making boxes of equiprobable volumes between 0 and 1.
- create a tree of depth three for three flips of a biased coin with bias p and q where the first flip is 1/2 heads and 1/2 tails and then the probabilities become p and q.

- Show that in this situation the probability of the event of getting a head on the 3rd trial is p.
- Prove (typically by induction) that the probability of the event of getting a head on the kth flip after flipping n time (k>1) is p. How about tails?
- Due Oct 29

- 3.16,3.17,3.18,3.21,3.33
- Consider the Markov Model of coin flip where the P(H|H)= a, P(H|T)=b, P(T|H)=1-a and P(T|T)=1-b. Suppose probability of a head on first flip = probability of a tail =1/2. Take the model to 3 flips.

- (graduate prob) Now "reverse" the model by starting with the third flip with the probabilities of getting a head on the third flip as calculated (same with the tail). Now calculate the conditional probability of getting a Head on the second flip given one on the third (same with all the combos) and make the reverse tree . What is similar and dissimilar about this reverse tree.
- If a=1/4 and b=1/2 find a probability for heads and tails on the first flip that will stay the same for succeeding flips (not 1/2)

- (graduate) what can you say about this tree?
- Due Nov 5

- quiz hw
- problems: 3.59, 3.62
- theoretical ex: 3.2, 3.9
- Show that if A,B,C are independent then any sequence of the three distinct sets or their negations are independent (ie: A~BC but do not use A~AC-- all the letters must be different)
- Show that A and B are independent given G iff P(A|BG)=PA|~BG).
- Test Nov 12
- Due Nov 20

- Ch4 problems 4.1,4.2,4.5,4.7,4.8,4.10
- Find a grapher that will graph the Poisson Distribution and graph it for lambda= .1,.5,10( we will discuss what you see)
- Due Nov 26

- Ch 5 problems 5.1,5.2,5.4
- Ch 4 problems 4.13,4.17,4.19,4.21,4.25,4.35
- Due Dec 10

- 4.37,4.38
- 5.6,5.7
- Due by Sun at 5 - will post solutions after that

- problems in pdf notes
- Create two random variables X,Y that you can prove are not independent.
- If Variance(X) = 10**(-6) (ten to the -6) and E(X)=0 what does Chebyshev say about the probability that |X|> 10**(-2). Could there possibly be a value of X >200. Explain.
- Explain in 5 lines or less why the Weak Law of Large Numbers tells you that E(X) is an important quantity.
- Let S be the number of heads in 3 Bernoulli trials (3 independent flips of a biased coin (p,q)). Show directly that the E(S)=3p and Var(S)=3pq. How do we get this result from our theorems about the expected value and the variance of the sum of random variables.
## Class Notes:

- Dec 3-- Expectaion,Variance, formulas with pdf, Independence of Random Variables.
- Dec 12 notes from ipad cleaned up but I have simpler proofs below.

Department of Mathematics and Computer Science, Lehman College, City University of New York