Statistics 414
Introduction to Probability Theory
Fall 2009

Tentative Course Outline

Course Outline

Textbook: Probability and Statistical Inference, seventh or eighth edition by R. W. Hogg and E. A. Tanis. All chapter and section numbers below refer to the seventh edition.

Week	Topics
1: Aug. 24-28	Basic concepts: random variables, sample space and events (with set notation), rules of probability. (1.1-1.2)
2: Aug. 31-Sept. 4	Combinatorics; conditional probability (1.3-1.4)
3: Sept. 7-11	No Monday class; Labor Day.Conditional probability, independence, Bayes' theorem(1.4-1.6)
4: Sept. 14-18	Bayes' theorem; discrete distributions; expectation, variance (1.6, 2.1-2.3)
5: Sept. 21-25	Moment generating functions; Examples: Bernoulli, Binomial, Poisson distributions (2.4-2.6). Others: Geometric, Negative Binomial, Hypergeometric.
6: Sept. 28-Oct. 2	Continuous distributions: density function, distribution function, quantiles, expectation, variance. (3.1,3.2)
Friday, Oct. 2: Midterm exam #1. This date is now fixed! The midterm will cover chapters 1 and 2.
7: Oct. 5-9	Continuous-type distributions: Densities, cdfs, mgfs, quantiles, expectations, and examples (3.3-3.5)
8: Oct. 12-16	normal distributions, bivariate distributions(3.6, 4.1)
9: Oct. 19-23	Multivariate probabilities involving nonrectangular support, covariance and correlation (4.1, 4.2)
10: Oct. 26-30	Conditional distributions, bivariate normal distributions (4.3, 4.4)
11: Nov. 2-6	Order statistics, functions of one random variable (8.3, 5.1 in 8th ed. or 6.9, 3.5 in 7th ed.)
Friday, Nov. 6: Midterm exam #2 This date is now fixed! The midterm will cover chapters 3 and 4 along with the section on order statistics
12: Nov. 9-13	Multivariate transformations, sums of independent random variables (5.2 - 5.4)
13: Nov. 16-20	The MGF technique, sample mean and sample variance (5.4, 5.5)
Nov. 23-27: No class; Thanksgiving break
14: Nov. 30-Dec. 4	Central limit theorem, approximations for discrete distributions (5.6, 5.7)
15: Dec. 7-11	Review
16: Dec. 15	Final exam: Tuesday, Dec. 15, 4:40-6:30pm, 358 Willard

Desired outcomes, day by day

Below is a list of skills students are expected to master in MATH/STAT 414, listed by the class(es) in which they were addressed.

If you click on each date's link, you will see the outlines used in class that day on the overhead projector.

Aug. 24: Students will be able to:
- identify the sample space of a probability experiment;
- define a random variable.
Aug. 26: Students will be able to:
- define a random variable and describe simple examples of RVs;
- apply set notation to simplify expressions involving events;
- apply Venn diagrams to the same.
Aug. 28: Students will be able to:
- prove simple facts based on the three axioms of probability;
- use set notation or Venn diagrams for certain probability problems.
Aug. 31: Students will be able to:
- use set notation or Venn diagrams for certain probability problems;
- exploit the multiplication principle when outcomes are equally likely;
- define a permutation on n elements and understand n! notation.
Sept. 2: Students will be able to:
- define a permutation on n elements and understand n! notation.
- recognize _nP_r and _nC_r situations; know formulas.
Sept. 4: Students will be able to:
- recognize multinomial coefficient situations and know the formula;
- define the conditional probability of A given B using a formula;
- understand intuitively what P(A | B) means;
- solve conditional probability problems using formulas or Venn diagrams.
Sept. 9: Students will be able to:
- state and use the multiplication rule of conditioning;
- state and use the definition of independent events.
Sept. 11: Students will be able to:
- state and use two forms of the Law of Total Probability;
- solve problems using Bayes' Theorem.
Sept. 14: Students will be able to:
- solve problems using Bayes' Theorem;
- define the c.d.f. of a random variable;
- recognize the difference between a continuous and discrete c.d.f.;
- define countable, discrete, and support;
Sept. 16: Students will be able to:
- define the p.m.f. of a discrete random variable;
- evaluate an unknown multiplicative constant in a p.m.f.
- recognize a hypergeometric r.v.;
- calculate the expectation of a (function of a) discrete r.v.
Sept. 18: Students will be able to:
- calculate the expectation of a (function of a) discrete r.v.
- explain what the linearity of the expectation operator means.
- define the mean, variance, and standard deviation;
- evaluate the mean, variance, and standard deviation for discrete random variables.
Sept. 21: Students will be able to:
- derive the moment generating function for a discrete r.v.;
- find E(X^r) for an arbitrary r using the m.g.f. of X;
- define a binomial(n,p) random variable.
Sept. 23: Students will be able to:
- define a binomial(n,p) random variable;
- use the binomial theorem;
- state the mean and variance of a binomial r.v.
Sept. 25: Students will be able to:
- state and use the p.m.f. of a Poisson r.v.;
- state the mean and variance of a Poisson r.v.
Sept. 28: Students will be able to:
- state and use the p.m.f. of a Poisson r.v.;
- state the mean and variance of a Poisson r.v.;
- define a negative binomial(r,p) random variable.
Sept. 30: Students will be able to:
- use the Poisson approximation to the binomial.
Oct. 5: Students will be able to:
- list the properties of a density function for a continuous r.v.;
- find the constant multiplier for a density by setting the integral to 1;
- give formulas for mean and variance of a continuous r.v.;
- differentiate a continuous c.d.f. to find a p.d.f.
- integrate a p.d.f. to find the c.d.f.
Oct. 7: Students will be able to:
- derive the m.g.f. of a continuous random variable;
- define "quantile" and related terms (percentile, quartile, etc.);
- evaluate quantiles of a given continuous distribution;
- give the p.d.f. and c.d.f. of a uniform random variable;
- give the p.d.f. and c.d.f. of an exponential random variable.
Oct. 9: Students will be able to:
- give the p.d.f. and c.d.f. of a uniform random variable;
- give the p.d.f. and c.d.f. of an exponential random variable;
- Understand the relationship between a Poisson process and an exponential distribution.
Oct. 12: Students will be able to:
- write the density function for a normal (mu,sigma²) random variable;
- derive the mean, variance, and m.g.f. of a normal random variable;
Oct. 14: Students will be able to:
- write the density function for a normal (mu,sigma²) random variable;
- derive the mean, variance, and m.g.f. of a normal random variable;
- understand a joint p.m.f. of two discrete random variables;
- derive a marginal p.m.f. from a joint p.m.f.
Oct. 16: Students will be able to:
- use a joint p.m.f. to check independence;
- give the joint mass function for a trinomial distribution;
- understand a joint density of two continuous random variables.
Oct. 19: Students will be able to:
- understand a joint density of two continuous random variables;
- set up and perform double integrals to find joint probabilities;
- derive a marginal density from a joint density.
Oct. 21: Students will be able to:
- describe the inversion method for generating random numbers;
- find the covariance of jointly distributed random variables.
Oct. 23: Students will be able to:
- find the correlation of jointly distributed random variables.
Oct. 26: Students will be able to:
- write the formula for a conditional density or p.m.f.;
- use a conditional density or p.m.f. like any other density or p.m.f.;
- find the mean or variance of one r.v. conditional on another.
Oct. 28: Students will be able to:
- derive the constant for a standard normal density;
- know where to look up the bivariate normal density;
- know the conditional distributions for a bivariate normal.
Oct. 30: Students will be able to:
- find distributions of minimum and maximum from a random sample.
Nov. 2: Students will be able to:
- find distributions of minimum and maximum from a random sample;
- use the c.d.f. method to find the distribution of a function of a r.v.
Nov. 4: We did not cover new material. Instead, we spent the class reviewing material for the upcoming midterm.
Nov. 9: Students will be able to:
- use the c.d.f. method to find the distribution of a function of a r.v.;
- use the change of variables method to find the distribution of a function of a r.v.;
- use the change-of-variables formula for a bivariate density function.
Nov. 11: Students will be able to:
- use the change-of-variables formula for a bivariate density function;
- write the joint density or p.m.f. of n i.i.d. random variables.
Nov. 13: Students will be able to:
- write the joint density or p.m.f. of n i.i.d. random variables;
- find expectation of the product of independent random variables;
- find the m.g.f. of the sum of independent random variables.
Nov. 16: Students will be able to:
- use formulas for mean and variance of sums of r.v.s;
- state the mean and variance of the sample mean;
- prove facts about sums of independent r.v.s using m.g.f.s;
- derive the m.g.f. of the sample mean from an i.i.d. sample.
Nov. 18: Students will be able to:
- define a chi-square distribution with r degrees of freedom and give its mean and variance;
- give the distribution of a linear combination of independent normals.
Nov. 20: Students will be able to:
- demonstrate that the sample variance has expectation sigma²;
- define a t distribution with r degrees of freedom.

Statistics 414Introduction to Probability TheoryFall 2009

Tentative Course Outline

Desired outcomes, day by day

Statistics 414
Introduction to Probability Theory
Fall 2009