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
Institutional Competencies Learning Outcomes
Analytical Thinking
The ability to reason, interpret, analyze, and solve problems from a wide array of authentic contexts.
