Stat 241 notes : fall 1997

Statistics 241/541: Notes (fall 97)

Chapter 1 : Probabilities and random variables
Sample spaces. Events. Rules for manipulating probabilities. Conditional probabilities. Conditional independence. Markov chains. Distributions of random variables. Geometric(p) distribution.

Chapter 2 : Expectations
Rules for (conditional) expectations. Expected number of tosses to tthh. Expected value for geometric(p). Coupon collector's problem. Sampling from an urn (analog of geometric). The problem of the big pills and little pills. Appendix to Chapter 2 on the interpretation of expectations as fair prices.

Chapter 3 : Things binomial
Binomial coefficients, binomial distribution (Bin(n,p)). Expected value of the binomial. How to get mugged. Bayes's formula. Rudimentary hypothesis testing. Digression on confidence intervals.

Chapter 4 : Variances and covariances
Definitions and basic facts. Uncorrelated versus independent. Probability bounds (Tchebychev inequality). Variance as a measure of concentration in sampling theory: comparison of sampling with and without replacement. Variances via conditioning.

Chapter 5 : Unexpected Symmetry
The Polya urn model (includes the hypergeometric distribution). Bet red: a question of optimal strategy. The Ballot Theorem.

Chapter 6 : Continuous distributions
Integral representation of Binomial tails. Uniform order statistics. Density functions. Beta function. Beta distribution. Expectation of a function of a random variable with a continuous distribution.

Chapter 7 : Normal distribution
De Moivre's approximation to the Binomial tails. Standardization. Standard normal density. Expected value and variance of N(mu,sigma^2). Approximation for coin tossing. Example with Hispanic counts on questionnaires. Central limit theorem. Boxplots. Appendix A: where the \sqrt{2\pi} comes from. Appendix B: Derivation of Stirling's formula. Appendix C: Derivation of de Moivre approximation via beta integral.

Chapter 8 : Poisson approximation
The Poison distribution and its properties. Poisson(np) approximation to the Bin(n,p). Poisson approximation with dependence, illustrated by matching problem. Sketch of the Chen-Stein method, as applied to the matching problem.

Chapter 9 : Poisson processes
The Poisson process as idealized-very-fast-coin-tossing. Distribution of the time to the first point. Gamma function and gamma distribution. Expected value for the gamma. Exponential distribution. Analogs between continuous and discrete time: gamma versus negative binomial; exponential versus geometric. Gamma(1/2). Poisson process of arrivals: superposition of independent processes, with coin tossing interpretation. New version: 28 October 97. Last two pages corrected.

Chapter 10 : Joint densities
Definition of jointly continuous distributions and joint densities. Joint densities for independent random variables. Joint densities from linear transformations, and smooth transformation (Jacobians). Example constructing beta from independent gammas. Beta function/gamma function identity. Sums of independent gammas, with chi-squared as special case. Appendix: Determinant formula for area of a parallelogram.

Chapter 11 : Conditional densities
Three related methods for calculating a conditional density, with conditioning on the value of a random variable with a continuous distribution.

Chapter 12 : Multivariate normal
Density for standard bivariate normal with correlation rho. Conditional distributions. Regression (to the mean). Rotation of axes and change of coordinates. Joint distribution of sample average and sample variance.

Chapter 13 : Generating functions
Probability generating functions for random variables taking (positive) integer values: identification of distributions and moments. Gamma mixture of Poisson gives negative binomial. Branching processes (including expansion of a 64th degree polynomial!). Moment generating functions. A little more on normal approximation to the binomial.

Material that never made it into the notes

Matlab m-files