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.

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.