Course Description
This course provides a rigorous foundation in probability. Topics include sample space, events, probability rules, discrete and continuous distributions, expectation, variance, moment-generating functions, multivariate distributions, and transformations of random variables. Emphasis is placed on foundational concepts essential for advanced statistical inference.
Athena Title
Math Statistics I
Prerequisite
MATH 2270 or MATH 2270H or MATH 2500 or MATH 2500E
Semester Course Offered
Offered fall and spring
Grading System
A - F (Traditional)
Student Learning Outcomes
- Students will define and explain fundamental concepts of probability, including sample spaces, events, conditional probability, and independence.
- Students will apply probability rules, including Bayes’ theorem, to solve problems involving conditional probabilities in real-world contexts.
- Students will analyze discrete and continuous probability distributions, including their expectations, variances, and moment-generating functions, to model different types of data.
- Students will compute and interpret probabilities and expectations for well-known distributions such as Binomial, Poisson, Normal, and Gamma, and justify their use in different scenarios.
- Students will evaluate and compare multivariate probability distributions, including marginal and conditional distributions, independence, and covariance between random variables.
- Students will derive and use probability distributions of functions of random variables using the method of transformations and Jacobians.
- Students will interpret and apply Tchebyscheff’s theorem to assess the dispersion of probability distributions and its implications in statistical inference.
Topical Outline
- Introduction
• Introduction to probability and statistical inference
- Probability
• Review of set notations
• Sample space, events, and probability of an event
• Conditional probability and Independence
• Bayes’ rule and applications
- Discrete Random Variable and Their Probability Distributions
• Expectation and Variance
• Well-known discrete distributions (e.g., Binomial, Geometric, Negative Binomial, Poisson, Hypergeometric) and their applications
• Moment generating functions
- Continuous Random Variables and Their Probability Distributions
• Expectation and Variance
• Well-known continuous distributions (e.g., Uniform, Normal, Gamma, Beta) and their applications
• Moment generating functions
• Tchebyscheff’s Theorem
- Multivariate Probability Distributions
• Bivariate and multivariate probability distributions
• Marginal and conditional probability distributions
• Independent random variables
• Expected value of a function of random variables
• Covariance of two random variables
- Functions of Random Variables
• Probability distribution of a function random variables
• Method of transformations using Jacobians
