Introduction to Probability and Statistics

Department: 
MATH
Course Number: 
3215
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every semester

This course is a problem oriented introduction to the basic concepts of probability and statistics, providing a foundation for applications and further study.

MATH 3215, MATH 3235, and MATH 3670 are mutually exclusive; students may not hold credit for more than one of these courses. 

Prerequisites: 

MATH 2401 or MATH 24X1 or MATH 2411 or MATH 2551 or MATH 2561 or MATH 2550 or MATH 2X51 or MATH 2605.

Course Text: 

At the level of Probability and Statistical Inference, Hogg and Tanis, 9th edition, Pearson

Topic Outline: 

Basic Principles Covered:

  • Multiplication principle, combinations, permutations
  • Inclusion-exclusion
  • Expected value, variance, standard deviation
  • Conditional probability, Bayes rule, partitions
  • Random variable, p.d.f., c.d.f., m.g.f.
  • Independence
  • Joint distributions, marginals, conditional expectations
  • Covariance, correlation
  • Transformations of a random variable
  • Central limit theorem, approximations
  • Basic distributions: uniform, binomial, multinomial, normal, exponential, Poisson, geometric, Gamma, Chi-squared, Student t, use of tables

Topics:

Probability

  • Experiments, events, sets, probabilities, and random variables
  • Equally likely outcomes, counting techniques
  • Conditional probability, independence, Bayes' theorem
  • Expected values, mean, variance, binomial and geometric distributions
  • Poisson, moment generating functions
  • Continuous random variables, exponential, gamma, and normal; intuitive treatment of the Poisson process and development of the relationship with the gamma distributions
  • Uniform and simulation
  • Multivariate distributions, calculation of probability, covariance, correlation, marginals, conditions
  • Distributions of sums of random variables
  • Central limit theorem

Statistics

  • Maximum likelihood, optimal, and unbiased estimators, examples
  • Univariate transformations using the chi square as an important example
  • Develop the idea of confidence intervals, confidence intervals for means with known variance in the normal case, large sample confidence intervals for means. Give the results of such a development for differences in the two-sample problem.
  • Confidence intervals for variances and ratios of variances and applied problems
  • Large sample confidence intervals for proportions, the one and two sample case, with examples and sample size considerations
  • Introduction to formal hypothesis testing, calculation of size and evaluation of the power function. One and two sample tests of hypotheses for normal means and variances. Chi-squared goodness of fit test.