# Math 3355 Probability

### Fall 2009

Last updated November 23, 2009.

### Professor James J. Madden

Course policies and syllabus.

IMPORTANT INFORMATION: Take home test to be distributed on Nov. 24. Due Nov. 30. See below.

### Schedule of lecture topics, links to notes and resources and homework assignments.

 Day and Date Section Topics Notes Homework Problems Assigned T 8/25 1.1, 1.2 Probabilistic Experiments Experiment, outcome, sample space event. Notes Due 9/1. 9: 4, 6 (assume different identity for each coin), 13, 20. Th 8/27 1.3, 1.4 Probability Axioms Practice problems. Rules for assigning probabilities to events Due 9/3. 23: 5, 7, 15, 25, 26, 27. 37: 14. T 9/1 2.1, 2.2, 2.3 Counting principles Matching problem. (Needs Mathematica; get student verion here). Due 9/8. 44: 8,17, 22, 27, 28. 50: 4, 5, 12, 24, 28(sol). 63: 1, 2, 3, 4, 5. Th 9/3 2.4 Combinations n choose r and applications. Poker hands. Due 9/8. 65: 20 (find the probability of each kind of poker hand). T 9/8 2.4, 3.1 Combinations (cont.), Conditional prob. Examples. Due 9/15. 63: 9(soln), 13, 29. 71: 3, 5, 6, 8, 21(soln). 82: 3, 4, 7, 13(soln). Some simple worked examples from Ch. 2 here. Th 9/10 3.1, 3.2, 3.3, 3.4 Conditional prob. and related ideas Summary of main formulae. 3.2 and trees. Bayes Theorem example (exercise 1) worked with Bayes and by 2-by-2 table, with marginals. Due 9/15: 82: 9(soln), 11, 17. 87: 6, 9. 96: 8, 19(soln), (22 for bonus pts). 106: 3, 15 (solution to 15). I recommend doing problems 1-10 in 3.1, 3.2, 3.3, and 3.4, but do not hand in. T 9/15 3.5 Independence Definitions; simple examples; two-by-two tables. Study! Th 9/17 3 Review Study! T 9/21 3.5 Test Test Test with Answers Th 9/23 4.1, 5.1 Random variables; binomial distribution Activity sheet Finish activity sheet. T 9/28 4.4, 5.1 binomial distribution (cont.); expectation. Properties of binomial distribution with parameters n and p. (Mathematica notebook on binomial distriction, here.) Definition of expectation. View some neat graphics related to the binomial distribution: Sampling from a population | Binomial and Normal. Problems on expectation: 173: 3, 6, 13. If I have n different pairs of socks in the dryer, each pair of a different color and design, and I take them out one sock at a time, how many socks should I expect to have removed when I first get a match? (Try this for n = 3, 4, 5.) ( The general answer is: (4^n)/Binomial[2 n, n].) (Also see: this blog.) T 10/6 4.5 Variance; mean of binomial. Proof of expectation of binomial 182: 6, 10. 186: 4, 5. 196: 6, 10, 19, 24. Th 10/8 4.5, 4.6, 5.1 Review of basic concepts: random variable, probability mass function, expectation. Handout. 199: 25. Problem: If you roll a single die repeatedly, how many rolls on average will it take to get a 6? If you roll repeatedly, how many rolls on average will it take to get a number strictly bigger than 4? If you roll repeatedly, how many rolls will it take on average to record all six possible numbers? T 10/13 Homework. Poisson distribution. Graph of class data. none assigned Th 10/15 Overview of ch 4 and 5 Concept Summary Due 10/20. From the handout on 10/8: 41, 44, 45, 48, 55. T 10/20 6.1, 6.2 Continuous distributions Examples. Definitions. PDF, CDF. Finding PDF of g(X). Due 10/27. 245: 1, 3, 5, 7. Th 10/22 6.3 Expectation Definition. Expressing expectation via CDF. E(g(X)). Examples (Cauchy distribution derived. When [0,1] divided at random point, what is expected length of the part containing p.) Due 10/27. 254: 1, 3, 5, 9. T 10/27 7.1 Uniform distribution Find the standardization of the uniform distribution on [a, b]. Th 10/29 7.2 Normal distribution Look at this cartoon of the binomial and the normal. TAKE HOME TEST DUE 11/3. (Extra credit problem included.) See here. T 11/03 5.2, 7.3 Poisson Processes and Exponential Distribution 212: 16, 19 Th 11/05 7.3 (cont) Exponential 290: 5, 8, 11 T 11/10 8.1 Joint Distributions 326: 5, 11, 13 Th 11/12 8.2 Independence 339: 1, 3, 6, 8, 9, 13 T 11/17 7.4 & 8.3 Gamma Distr. & Conditional Distr. Prove that Gamma(1)=1, Gamma(r+1) = r Gamma(r) (cf. p. 293), so Gamma(n+1) = n!. 296: 2, 6, 8. Th 11/19 8.3 & 8.4 Conditional Expectation. Sums of independent variables and convolution. Quiz. Study 212: 19, 290: 11, 296: 3. 355: 11,12, 21 T 11/24 Quiz. Similar to the last problem (on normal distributions) from this old final. Also, you will receive a take-home test.