STA341 Statistical Theory I

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COURSE DESCRIPTION

This course provides a calculus-based treatment of probability, which forms the foundation of statistics. Students study probability theory, combinatorics, random variables, discrete and continuous distribution theory, expected values, moment-generating functions, multivariate distributions, functions of random variables, and conditional and marginal probability distributions, and the Central Limit Theorem.

GENERAL COURSE GOALS

To have students:
  • Expand their understanding of the nature and history of mathematics.
  • Learn to use mathematics to gain insight into areas of application, for example in the physcial and life sciences, and economics.
  • Appreciate the importance of mathematics in the development and advancement of human civilization.
  • Learn the appropriate use of technology including how to use it and when to be skeptical of its results.
  • Develop critical thinking and a basic understanding of logic essential for lifelong learning.

SPECIFIC COURSE OBJECTIVES

For students:
  • To understand the concepts of set-based probability theory, combinatorics, random variables, discrete and continuous distribution theory, moment-generating functions, functions of random variables, and conditional and marginal probability distributions.
  • To be able to apply probability to real-life applications.
  • To learn the historical development of the field of probability.
  • To approach problems algebraically, geometrically, numerically, and symbolically.

LEARNING OUTCOMES

By successfully completing this course, students should be able to:
  • Prove simple propositions in set theory
  • Define sample spaces and events for random experiments, using set notation
  • Solve counting problems involving permutations, combinations, and distinguishable permutations, knowing the difference between sampling with replacement and without replacement
  • State the axioms of probability and basic theorems on probability
  • Prove simple theorems involving probabilities, making use of set-notation and basic probability axioms and laws
  • Define and solve problems using conditional probability
  • Recognize the difference between independent and mutually exclusive events
  • Describe and apply the Total Probability Law and use it to derive Bayes Theorem.
  • Apply Bayes Theorem, when appropriate
  • Define random variables, both of the discrete type and the continuous type
  • Define probability mass functions and probability density functions, and to determine if a given function is a p.m.f. or a p.d.f.
  • Define a cumulative distribution function and its relation to p.m.f.s and p.d.f.s.
  • Compute probability, given a probability density or mass function
  • Compute probability, given a cumulative distribution function
  • Define and find mathematical expectation
  • Define and find mean, variance, and standard deviation.
  • Describe the essential features and application of the discrete distributions: binomial, hypergeometric, geometric, negative binomial, and Poisson,
  • Describe the essential features and application of the continuous distributions: uniform, exponential, gamma, normal, chi-square,
  • Define moment generating functions and their relation to mean, variance, and p.m.f.s
  • Determine distributions of functions of random variables (simple cases)
  • Describe distributions of more than one variable, and be able to solve problems involving functions of two variables (involves knowing when to sum and when to integrate. Ability to evaluate multiple integrals is needed)
  • Describe and solve problems involving correlation coefficient, conditional distributions
  • Know and apply basic theorems concerning several independent random variables, especially those concerning sums of independent random variables
  • State history of normal distribution, briefly.
  • State and apply the Central Limit Theorem
  • Describe how computers can be used to estimate probability and to simulate random samples from given distributions, using simulation via pseudorandom number generation.

SUPPORT OF THE UNIVERSITY MISSION

This course supports the mission of the university as follows: "...through teaching ... prepares men and women for responsible lives by imparting and expanding knowledge, developing skills, and cultivating enduring values. ... students develop their abilities for thinking clearly and creatively, enhance their capacity for sound judgment, and prepare for the challenge of learning throughout their lives."

PREREQUISITES

MTH 203 or equivalent (may be taken concurrently)

WHERE AND WHEN

  • Secion 01 : Course meeting time/place: T, Th, 5:00 P.M. - 6:15 A.M. Location: McAuley 206.

TEXTBOOK AND MATERIALS

  • Required: Probability and Statistical Inference" 9th Edition by Robert V. Hogg annd Elliot A. Tanis. Prentice Hall. ISBN-10: 0-321-58475-9, ISBN-13: 978-0-321-58475-5.
  • Strongly recommended: TI-83Plus or TI-84Plus graphing calculator. (Note: If calculators will be allowed on examinations, only the following models will be permitted: TI-83, TI-83Plus, TI-84Plus.)
  • Strongly recommended: Student Edition of MATLAB. (MATLAB will be used in various classroom demos and some problems on assignments will require the use of MATLAB.)

TEACHING METHODOLOGY

Lecture, question and answer, readings, homework exercises.

GENERAL REMARKS AND ADVICE

This web site is designed to, among other things, make it clear what is expected from you and what you can expect from the course and from me. This course will be challenging; it is fast-paced, it requires a great deal of meticulous attention to detail as well as imagination and creativity, and just about everything in it depends on your understanding of everything else in the course that preceded it. Nevertheless, if you work hard, do not allow yourself to fall behind, and seek help when you need it, you should be successful in this course.

Reading assignments and homework problems for nearly each class meeting are posted on the course web site. It is your responsibility to check the web site frequently (i.e., at least twice a day) for the homework. It is extremely important that you complete the reading assignments and try the homework problems before the following class meeting.

Last modified: 9/7/2017