Topics Covered in Lecture
All reading assignments are from the textbook "Probability" by Pitman.| Lect # | Date | Topics | Reading |
|---|---|---|---|
| 1 | 1/22 | Course logistics [Slides]; probability as proportions; probability interpretations; events and sets; partition; probability distribution; examples of probability distributions. | 1.1 - 1.3 |
| 2 | 1/24 | Conditional probability; multiplication rule; average conditional probabilities; independence of two events; Bayes' rule; prior probability; posterior probability. | 1.4, 1.5 |
| 3 | 1/29 | Sequence of independent Bernoulli trials; geometric distribution; sequence of dependent events; birthday problem; independence of n events, for n > 2; pairwise independence. | 1.6 |
| 4 | 1/31 | More on independence; sequence of Bernoulli trials; binomial distribution; consecutive odds ratio; Stirling's formula; mean and mode of Binomial(n,p). | 2.1 |
| 5 | 2/5 | Normal approximation of the binomial distribution; cumulative distribution; square root law; confidence interval. | 2.2 |
| 6 | 2/7 | More on the binomial distribution: demonstration of convergence to the Normal distribution, empirical distribution, law of large numbers, and convergence to the Poisson distribution; more on the normal approximation; Poisson approximation; ball and bins. [ R script ] [ README ] [ The R Project ] | 2.2., 2.4 |
| 7 | 2/12 | Random sampling with or without replacement; hypergeometric distribution and multivariate generalization; elementary combinatorics. | 2.5 |
| 8 | 2/14 | Random variables; joint distributions; marginal distributions; identical distributions; equality of random variables; conditional distribution of a random variable; independent random variables. | 3.1 |
| 9 | 2/19 | Expectation; linearity of expectation; indicator r.v.; expected count; tail sum formula for expectation; exchangeability | 3.2, 3.6 |
| 10 | 2/21 | Variance; Markov's inequality; Chebychev's inequality; weak law of large numbers | 3.3 |
| 11 | 2/26 | Central limit theorem; infinite discrete distribution; geometric distribution; infinite sum rule; expectation of an infinite discrete random variable; Riemann rearrangement theorem; waiting time to the jth success; negative Binomial distribution | 3.4 |
| 12 | 2/28 | Coupon collector's problem; properties of Poisson random variables; sum of independent Poisson random variables; random scatter; Poisson scatter theorem | 3.5 |
| 13 | 3/5 | More on symmetry and exchangeability; the mean and variance of a Hypergeometric(N,M,n) random variable | 3.6 |
| 14 | 3/7 | More on exchangeability; continuous random variable; probability density function; continuous uniform distribution. | 4.1 |
| 15 | 3/12 | Midterm | |
| 16 | 3/14 | The memoryless property of geometric r.v.s; continuous limit; exponential random variables | 4.2 |
| 17 | 3/19 | Equivalent characterizations of a sequence of independent Bernoulli(p) trials; Poisson arrival process; sum of equidistributed exponential r.v.s. (Gamma distribution); change of variables for 1-to-1 functions of random variables; uniform distribution over the 2-sphere. | 4.4 |
| 18 | 3/21 | Change of variables for many-to-1 functions of random variables; cumulative distribution function; uniform process; order statistics; the c.d.f and the density of the kth order statistic; the distribution of gaps in the uniform process; order statistics for general i.i.d. random variables | 4.5, 4.6 |
| 3/26 | Spring Recess | ||
| 3/28 | Spring Recess | ||
| 19 | 4/2 | Minimum and maximum of independent random variables; Recapitulation of the uniform process; introduction to continuous joint distribution; uniform distribution in 2-dimensions; the probability that gaps L1, L2, L3 in the uniform process with n=2 form a triangle; n-dimensional Euclidean ball. | 5.1 |
| 20 | 4/4 | Joint probability density function of a pair of random variables; marginal densities; density of independent r.v.s; competing independent exponential r.v.s; joint density of a pair of order statistics; distribution of gaps in the uniform process | 5.2 |
| 21 | 4/9 | Beta distribution; Dirichlet distribution; conjugate prior; n-ball and n-sphere; independent normal distributions; Rayleigh distribution; derivation of the chi(n) distribution; derivation of the chi2(n) distribution. | 5.3 |
| 22 | 4/11 | Pearson's chi2 test of goodness-of-fit; p-value; convolution of functions (commutativity and associativity); applications of the convolution formula | 5.4 |
| 23 | 4/16 | Introduction to Markov chains; transition probability; 1-dimensional random walk with absorbing boundaries at 0 and N; first-step analysis; probability of hitting 0 before hitting N (gambler's ruin); conditional expectation; the tower property; expected time to hitting either 0 or N. NOTE: Markov chains are not discussed in the textbook. If you missed today's class, try to get the lecture note from a friend. |
6.1 - 6.2 |
| 24 | 4/18 | Derivation of the probability of hitting 0 before N in a 1-d random walk; continuous versions of the following concepts: conditional probability, Bayes' Rule, conditional expectations; conditional density; law of successive conditioning. | 6.3 |
| 25 | 4/23 | Covariance; Pearson's correlation; scale-free property; Cauchy-Schwarz inequality; correlation of type sizes in multinomial sampling | 6.4 |
| 26 | 4/25 | Bivariate normal (X1,X2): construction from independent N(0,1) random variables; joint, marginal, and conditional densities; the distribution of a X1 + b X2 + c; covariance matrix; multivariate normal distribution. | 6.5 |
| 27 | 4/30 | Ordinary and exponential generating functions for a sequence of real numbers; generating function for a convolution of sequences; probability generating function; p.g.f. for a sum of independent random variables; applications in random walks. | |
| 28 | 5/2 | The sum of a random number of i.i.d. random variables, and its generating function, mean, and variance; branching process; the mean and the variance of the population size at time t; extinction probability. | |
| 5/16 | Final Exam: 8:00am-11:00am in 100 GPB |