Honors Probability and Statistics

Department: 
MATH
Course Number: 
3225
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every other fall semester

The topics covered parallel those of MATH 3215, with a more rigorous and intensive treatment. Credit is not allowed for both MATH 3215 and 3225.

Prerequisites: 

You must have B or higher in MATH 2401, MATH 2411, or MATH 2605 or MATH 2551 or MATH 2561 to be enrolled in this course.  If you are in one of the above courses in the spring or summer term, you may register for MATH 3225 during Phase I of registration. In early August, we will drop any students in 3225 (fall term) who make C, D, or F in 2401, 2411, 2605, 2551 or 2561 in the spring or summer term.

Course Text: 

At the level of Probability and Statistical Inference, Hogg and Tanis, 8th edition

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
  • Simulation, transformations of a random variable
  • Central limit theorem, approximations
  • Basic distributions: uniform, binomial, multinomial, normal, exponential, Poisson, geometric

Topics:

  • 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
  • Univariate transformations using the chi square as an important example
  • Distributions of sums of random variables including convolution techniques
  • Central limit theorem
  • Develop the chi square and review the Gamma and its relationship with the Poisson process
  • Introduce the t and F distributions and some of their properties, including the use of tables. The form of the density need not be derived but the relationships with the normal and chi-square should be developed.
  • Computation of mles for some standard examples
  • Role of variance for unbiased estimators and the use of the Cramer-Rao lower bound
  • Develop the idea of confidence intervals, confidence intervals for means with known variance in the normal case, large sample confidence intervals for means, and small 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.