Calculus Based Probability

A Study of Some Named Distributions

What you'll learn
- Be able to use basic counting techniques (multiplication rule, combinations, permutations) to compute probability and odds;
- Be able to compute conditional probabilities directly and using Bayes’ theorem, and check for independence of events
- Be able to understand the definition of a random variable and its associated functions
- Be able to set up and work with discrete random variables; in particular, to understand the Bernoulli, binomial, geometric, Poisson, and other distributions;
- Be able to work with continuous random variables. In particular, know the properties of uniform, normal, and exponential distributions
- Be able to work with joint distributions of both discrete and continuous random variables
- Understand the central limit theorem
- Be able to compute confidence intervals for population parameters

Requirements
- Calculus and Multivariable Calculus

Description
This course offers a rigorous introduction to the mathematical foundations of probability and statistics, emphasizing applications across the natural, physical, and social sciences. Students will explore the axioms of probability theory, combinatorial analysis, and fundamental concepts such as sample spaces and events. Key topics include conditional probability, Bayes’ theorem, and the law of total probability, providing a framework for understanding complex probabilistic scenarios.

The curriculum delves into discrete and continuous random variables, examining distributions such as binomial, geometric, Poisson, exponential, uniform, and normal. Students will learn to compute expectations, variances, and apply the law of the unconscious statistician for function transformations. The course also covers joint, marginal, and conditional distributions, covariance, and independence, with an introduction to moment-generating functions.

A significant focus is placed on the Central Limit Theorem and its implications for statistical inference. Students will engage with real-world applications, enhancing their analytical skills and preparing them for advanced studies in statistics, data science, and related fields. A solid understanding of Calculus II is required, and familiarity with multivariable calculus is beneficial for topics involving joint continuous distributions.

Students with minimal calculus backgrounds can still benefit from many early topics and especially discrete distributions.  One can still learn the underlying theory of probability without a background in Calculus

Who this course is for:
- math majors who want to supplement their course
- Individuals with a caculus background who want to self study
- Actuarial students preparing for the P-exam
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