#### Math 461 Probability Fall 2006 Class Log

 1. Aug 23, Wed. course overview and logistics; probability and randomness; examples from the course: 2 dice, birthday problem, cancer test, discrete and continuous probability distributions 2. Aug 25, Fri. Sec 1.2-5: multiplication principle, permutations, multinomial coefficients, binomial coefficients, combinations, examples 3. Aug 28, Mon. more on binomial coefficients; basic properties; Pascal's identity; committee-chairperson identity; combinatorial vs algebraic proofs; binomial thm; proof by induction 4. Aug 30, Wed. combinatorial proof of binomial theorem; multinomial theorem; alternate interpretation of multinomial coefficients; Sec 1.6: putting indistinguishable balls into urns; counting integral solutions of linear equations; Sec 2.2: sample space and events 5. Sep 1, Fri. Sec 2.3-4: axioms of probability and basic consequences; principle of inclusion-exclusion Sep 4, Mon. Labor Day - no class 6. Sep 6, Wed. Sec 2.5: probability spaces with equally likely outcomes; examples; matching problems 7. Sep 8, Fri. more on ordered vs unordered sample spaces; Sec 3.2: conditional probability; definition and examples 8. Sep 11, Mon. example of simpler calculations using conditional probability; Sec 3.3: Bayes' formula; Monty Hall problem 9. Sep 13, Wed. hiker problem using Bayes' formula; Sec 3.4: def of independent events; Bernoulli trials 10. Sep 15, Fri. In 239 Altgeld for introduction to Iprom 11. Sep 18, Mon. more on Bernoulli trials; problem of the points; gambler's ruin 12. Sep 20, Wed. probabilistic method; 2 examples: Ramsey theory and partitioning to split families of k-subsets 13. Sep 22, Fri. Sec 4.1-2: random variables; probability mass function; cumulative distribution function; coupon collector problem 14. Sep 25, Mon. Sec 4.3-4: expected value; Sec 4.5: variance 15. Sep 27, Wed. linearity of expectation; variance of linear transformation; Sec 4.6: Bernoulli and binomial random variables; expectation and variance; average power of voters; Stirling's approximation 16. Sep 29, Fri. Test 1 in class 17. Oct 2, Mon. Sec 4.7: Poisson approximation to Binomial random variables; expectation and variance; examples 18. Oct 4, Wed. Sec 4.8: geometric, negative binomial, and hypergeometric random variables; expectation and variance 19. Oct 6, Fri. Sec 5.1-5.2: continuous random variables; expectation; variance 20. Oct 9, Mon. binomial approximation to hypergeometric random variable; Sec 5.3: uniform continuous random variables; Sec 5.4: density function for normal distribution 21. Oct 11, Wed. expectation and variance of normal distribution; cumulative distribution function; approximation to binomial random variables 22. Oct 13, Fri. opinion polling; Sec 5.5: exponential random variables; expectation and variance; memoryless property 23. Oct 16, Mon. failure rates; Sec 5.6.1: Gamma distribution; Gamma function; interpretation 24. Oct 18, Wed. expectation and variance of Gamma distribution; Sec 5.6.3: Cauchy distribution; expectation and variance; Sec 5.7: densities of functions of random variables; example of second moment; informal early feedback 25. Oct 20, Fri. informal early feedback summary; Sec 6.1: joint distributions, discrete and continuous; multinomial distribution 26. Oct 23, Mon. examples; Sec 6.2: independent random variables; joint mass function when discrete; example 27. Oct 25, Wed. choosing k-subsets of n elements uniformly on a computer; joint density functions of continuous variables; example 28. Oct 27, Fri. Sec 6.3: sums of independent random variables; discrete, continuous, and convolution; sum of binomials, poissons, exponentials 29. Oct 30, Mon. sum of normals; relation to Fourier transform; Sec 6.4: conditional random variables: discrete case; 2 examples 30. Nov 1, Wed. Sec 6.5: conditional random variables: continuous case; example; mixed conditional and discrete conditional distributions 31. Nov 3, Fri. Test 2 in class 32. Nov 6, Mon. Sec 6.6: min and max of two random variables; Sec 7.2: properties of expectation; expectation of sums of random variables; examples 33. Nov 8, Wed. QuickSort complexity; random walk; example; Sec 7.3: higher moments of random variables that count 34. Nov 10, Fri. example; Sec 7.4: variance of sum of random variables; covariance; properties; sample mean and sample variance 35. Nov 13, Mon. correlation; multinomial distribution; Sec 7.5.1: conditional expectation; Sec 7.5.2: computing expectations using conditional expectations 36. Nov 15, Wed. proof of Prop 5.1; examples; Sec 7.5.3: computing probabilities when conditioning on a random variable 37. Nov 17, Fri. example; Sec 7.5.4: conditional variance; example; Sec 7.6: conditional expectation as a predictor Nov 20, Mon. -- Nov 24, Fri. Thanksgiving break - no class 38. Nov 27, Mon. Sec 7.7: moment generating functions; extracting higher moments; calculations for common random vars; mgf's for sums of independent random variables 39. Nov 29, Wed. more examples of mgf's; Sec 8.2: Markov inequality; Chebyshev inequality; Weak Law of Large Numbers 40. Dec 1, Fri. Sec 8.4: Strong Law of Large Numbers; use of Markov inequal to prove "almost all" statements 41. Dec 4, Mon. threshold functions; use of Chebyshev inequality; Sec 8.3: Central Limit Theorem 42. Dec 6, Wed. proof of Central Limit Theorem; Sec 9.2: Markov chains; definition; finite state Markov chains; transition probabilities; transition matrix; computing distribution of X_t; evaluations 43. Dec 8, Fri. absorbing, ergodic, and regular chains; stationary distribution; absorption probability Dec 14, Thurs., 8-11am Final Exam

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