Statistics 414
Introduction to Probability Theory
Spring 2009

Tentative Course Outline

Course Outline

Final exam summary sheet

Textbook: Probability and Statistical Inference, seventh edition by R. W. Hogg and E. A. Tanis

Courtesy of Professor Haran: Rough schedule for Stat 414, subject to change (check back often!) Sections refer to the Hogg and Tanis textbook.

Week	Topics
1: Jan. 12-16	Basic concepts: random variables, sample space and events (with set notation), rules of probability, simple combinatorics. (1.1-1.3)
2: Jan. 19-23	No Monday class; MLK Jr. Day. Combinatorics; conditional probability (1.3-1.4)
3: Jan. 26-30	Conditional probability, independence, Bayes' theorem(1.4-1.6)
4: Feb. 2-6	Discrete distributions; expectation, variance, moment-generating functions (2.1-2.3, 2.5)
5: Feb. 9-13	Examples: Bernoulli, Binomial, Poisson distributions (2.4,2.6). Others: Geometric, Negative Binomial, Hypergeometric.
6: Feb. 16-20	Continuous distributions: density function, distribution function, quantiles, expectation, variance. (3.1,3.2)
Friday, Feb. 20: Midterm exam #1. This date is now final; the midterm will cover chapters 1 and 2.
7: Feb. 23-27	Examples: Uniform, exponential, gamma, chi-square. transformations of random variables (3.1-3.4, 3.5)
8: Mar. 2-6	transformations of random variables, normal distributions, bivariate distributions(3.5, 5.2, 4.1)
Mar. 9-13: No class; Spring break
9: Mar. 16-20	conditional distributions, multivariate probabilities involving nonrectangular support (4.1, 4.3)
10: Mar. 23-27	Correlation coefficient, transformations of multivariate random variables (4.2,4.4)
11: Mar. 30-Apr. 3	Multivariate transformations, Independent random variables (4.4, 4.5)
Friday, Apr. 3: Midterm exam #2 This date is now final; the midterm will cover chapters 3 and 4.
12: Apr. 6-10	Order statistics, sums of independent random variables, Chebyshev's inequality (6.9, 4.6, 4.7)
13: Apr. 13-17	Central Limit Theorem, Approximations for discrete distributions (5.4, 5.5)
14: Apr. 20-24	Limiting moment-generating functions, Bivariate Normal distribution (5.7, 5.6)
15: Apr. 27-May 1	Review
16: May 4-8	Final exam: Monday, May 4, 12:20-2:10pm, 112 Walker Building

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.

Jan. 12: Students will be able to:
- identify the sample space of a probability experiment;
- define a random variable and describe simple examples of RVs.
Jan. 14: Students will be able to:
- apply set notation to simplify expressions involving events;
- apply Venn diagrams to the same;
- prove simple facts based on the three axioms of probability.
Jan. 16: 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.
Jan. 21: 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;
- recognize _nP_r and _nC_r situations; know formulas.
Jan. 23: Students will be able to:
- recognize _nP_r and _nC_r situations; know formulas.
- 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.
Jan. 26: Students will be able to:
- 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;
- state and use the multiplication rule of conditioning;
- state and use the definition of independent events.
Jan. 28: Students will be able to:
- state and use the multiplication rule of conditioning;
- state and use the definition of independent events;
- state two forms of the Law of Total Probability;
- solve problems using Bayes' Theorem.
Jan. 30: Students will be able to:
- solve problems using Bayes' Theorem;
- define countable, discrete, and support;
- define the p.m.f. of a discrete random variable;
- evaluate an unknown multiplicative constant in a p.m.f.
Feb. 2: Students will be able to:
- 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.;
Feb. 4: Students will be able to:
- 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.
Feb. 6: Students will be able to:
- derive the moment generating function for a discrete random variable;
- find E(X^r) for an arbitrary r using the m.g.f. of X;
- use the binomial theorem;
- define a binomial (n,p) random variable.
Feb. 9: Students will be able to:
- define Bernoulli trial and binomial (n,p) r.v.;
- write the p.m.f. of a binomial (n,p) r.v.;
- recognize disguised binomial r.v. problems;
- define the cumulative distribution function for any r.v.;
Feb. 11: Students will be able to:
- state the mean and variance of a binomial r.v.
Feb. 13: 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.
Feb. 16: Students will be able to:
- use the Poisson approximation to the binomial;
- 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.
Feb. 18: Students will be able to:
- give formulas for mean and variance of a continuous r.v.;
- integrate the p.d.f. of a continuous r.v. to find the c.d.f.;
- differentiate the c.d.f. of a continuous r.v. to find the p.d.f.
Feb. 23: Students will be able to:
- derive the m.g.f. of a continuous random variable;
- 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;
- define quantile and related terms (percentile, quartile, etc.);
- evaluate quantiles of a given continuous distribution;
Feb. 25: Students will be able to:
- understand the relationship between a Poisson process and exponential wait times;
- answer questions about counts or wait times for a Poisson process;
- use the c.d.f. method to find the distribution of a function of a random variable.
Feb. 27: Students will be able to:
- use the c.d.f. method to find the distribution of a function of a random variable.
- use the change of variables method to find the distribution of a function of a random variable.
Mar. 2: Students will be able to:
- use the c.d.f. method to find the distribution of a function of a random variable.
- use the change of variables method to find the distribution of a function of a random variable.
- generate random numbers using the inversion method;
- write the density function for a normal (mu,sigma²) random variable.
Mar. 4: Students will be able to:
- write the density function for a normal (mu,sigma²) random variable.
- recognize a binomial distribution problem in disguise;
- derive the mean and variance of a normal random variable;
- write Var(aX+b) in terms of Var(X).
Mar. 6: Students will be able to:
- derive the m.g.f. of a normal random variable;
- write M_aX+b(t) in terms of M_X(t);
- understand a joint p.m.f. of two discrete random variables;
- derive a marginal p.m.f. from a joint p.m.f.;
- use a joint p.m.f. to check independence.
Mar. 16: Students will be able to:
- understand and use a joint p.m.f. of two discrete random variables;
- give the joint mass function for a trinomial distribution;
- understand a joint density of two continuous random variables;
Mar. 18: 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;
- find the covariance of jointly distributed random variables.
Mar. 20: Students will be able to:
- find the covariance of jointly distributed random variables;
- find the correlation of jointly distributed random variables.
Mar. 23: Students will be able to:
- find a correlation given a joint p.m.f. or p.d.f.
Mar. 25: 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.;
Mar. 27: Students will be able to:
- use a conditional density or p.m.f. like any other density or p.m.f.;
- find a conditional mean or variance of one r.v. conditional on another;
- use the change-of-variables formula for a bivariate density function.
Mar. 30: Students will be able to:
- use the change-of-variables formula for a bivariate density function.
- understand terminology like "i.i.d." and "simple random sample";
- write the joint density or p.m.f. of n i.i.d. random variables.
Apr. 1: Students will be able to:
- find the expectation of the product of independent random variables;
- find the m.g.f. of the sum of independent random variables.
Apr. 6: Students will be able to:
- use conditioning to find joint expectations and probabilities;
- find distributions of the maximum from a random sample.
Apr. 8: Students will be able to:
- find distributions of the minimum and maximum from a random sample;
- use formulas for mean and variance of sums of r.v.s;
- state the mean and variance of the sample mean.
Apr. 10: Students will be able to:
- state the mean and variance of the sample mean;
- define convergence in probability of a sequence of r.v.s;
- state Chebyshev's inequality;
- state the Weak Law of Large Numbers;
- use Chebyshev's inequality to prove the finite-variance version of the WLLN.
Apr. 13: Students will be able to:
- define a chi-square distribution and give its mean and variance.
Apr. 15: 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;
- demonstrate that the sample variance has expectation sigma².
Apr. 17: Students will be able to:
- state the central limit theorem;
- use the CLT to approximate probabilities.
Apr. 20: Students will be able to:
- state the central limit theorem;
- describe roughly how to prove the CLT.
Apr. 22: Students will be able to:
- recognize when to apply the normal approximation to the binomial or the Poisson;
- find probabilities using these normal approximations;
- understand the continuity correction in these cases;
Apr. 24: Students will be able to:
- use and understand the normal approximation to the Poisson;
Apr. 27: Students will be able to:
- justify the Poisson approximation to the binomial using m.g.f.s.;
- characterize the bivariate normal distribution;
- know where to look up the bivariate normal density.
Apr. 29: Students will be able to:
- know where to look up the bivariate normal density;
- know the conditional distributions for a bivariate normal.

Statistics 414Introduction to Probability TheorySpring 2009

Tentative Course Outline

Desired outcomes, day by day

Statistics 414
Introduction to Probability Theory
Spring 2009