Probability Theory - Calculus-Based Statistics - Online Course for Academic Credit

Often referred to as the "higher Probability & Statistics course", or even "Calculus-based Statistics", our Probability Theory course is actually an introduction to the study of statistics and probability, but based upon the usage of Calculus to study both discrete and continuous aspects of the subject. Accordingly, there is no prerequisite of a previous study of statistics, but rather a prerequisite of having completed Calculus II (having completed or concurrently enrolled in Multivariable Calculus is recommended). The Probability Theory course is an essential part of the mathematical training for those wishing to study Data Science.

Completion of DMAT 311 - Computational Probability Theory earns 3 academic credit semester hours with an official academic transcript from Roger Williams University, in Providence, Rhode Island, USA, which is regionally accredited by the New England Commission of Higher Education (NECHE), facilitating transfer of credits nationwide to other colleges and universities.

Probability Theory Introductory Videos

Probability Theory Course Introduction

The curriculum for the course, Prob/Stat&Mathematica by Carpenter/Davis/Raschke/Uhl, is a thorough and advanced investigation of the subject matter, fascinating and challenging at the same time. The usage of the powerful computer algebra and graphing system Mathematica™ allows for a unique exploration of distributions - both discrete and continuous - and their application to the cornerstone of the subject - the data set from a real-world situation.

Probability Theory differs from the "lower" Statistics course significantly in both approach and difficulty level. Compare the prerequisites:

Roger Williams University Course Catalog Listing: DMAT 311 - Computational Probability Theory

DMAT 311 - Computational Probability Theory - Learning Outcomes

• 1. To understand the core concepts of Probability, Sampling, Distributions, and Density
• 2. To understand and compute Monte-Carlo method for integration
• 3. To understand and compute Expected Value, Variance, Mean, Mode, Median
• 4. To understand and compute Probability and Conditional Probability
• 5. To understand and compute Markov's and Chebyshev's Theorems
• 6. To understand and compute normally and exponentially distributions
• 7. To understand and compute calculus-based formulas and relationships between Cumulative Distribution Functions and Probability Density Functions
• 8. To understand and compute the Central Limit Theorem
• 9. To understand the core concepts of discrete and random variables
• 9. To understand and compute Joint Distributions, Correlations, and Covariance
• 10. To understand the Law of Total Probability
• 11. To understand and compute the classic statistics measures of Confidence Intervals and Hypothesis testing.

DMAT 311 - Computational Probability Theory - Syllabus of Topics

1.	Getting Started
	1.1.	Email and Chat
	1.2.	Learning About the Course
	1.3.	Required Hardware
	1.4.	Software Fundamentals

2.	Simulations
         2.1.	Uniform Distributions
         2.2.	Monte Carlo Methods
         2.3.	Random Walks
         2.4.	Shooting craps;  Iterated Fractals.

3.	Data Analysis
         3.1.	Frequency
         3.2.	Expected Value
         3.3.	Cumulative Distributions
         3.4.	Variance
         3.5.	Histograms
         3.6.	Related formulas for Expected Values and Variance

4.	Probabilities
         4.1.	Calculating Probability
         4.2.	Union and Intersection and Probability
         4.3.	Conditional Probability Formula
         4.4.	Independence
         4.5.	Indicator functions

5.	More Data Analysis
         5.1.	Markov's Inequality
         5.2.	Chebyshev's Theorem
         5.3.	Laws of Large Numbers
         5.4.	One-Sided Chebyshev Theorem

6.	Normal and Exponential Distributions
         6.1.	Approximately Normally Distributed Sets
         6.2.	Normal Distribution
         6.3.	Approximately Exponentially Distributed Sets
         6.4.	Exponential Distribution
         6.5.	Memoryless Property of Exponential Distributions

7.	Random Variables
         7.1.	“Random Variables” 
         7.2.	Discrete Random Variables
         7.3.	Continuous Random Variables
         7.4.	Probability Density Functions
         7.5.	Cumulative Distribution Functions
         7.6.	Expected Values and Variance
         7.7.	Markov, Chebyshev, and Law of Large Numbers Revisited
         7.8.	Mean, Median, and Mode

8.	Joint Distributions
         8.1.	Joint Probability Calculations
         8.2.	Discrete & Continuous
         8.3.	Expected Values, Covariance, and Correlation.
         8.4.	Conditional Probability Calculations
         8.5.	Conditional Expectations
         8.6.	The Law of Total Probability

9.	Central Limit Theorem
         9.1.	Generating Functions for Discrete Random Variables
         9.2.	Generating Functions for Continuous Random Variables
         9.3.	Generating Functions and Independence
         9.4.	Central Limit Theorem
         9.5.	Chi-squared and Gamma random variables

10.	Classic Statistics
         10.1    Sampling
         10.2.   Confidence Intervals
         10.3.   Hypothesis testing

Legacy Course Connection

Legacy Distance Calculus Course:
DMAT 315 - Probability Theory

Time commitments are important for success in an online Probability Theory course for college credit from Distance Calculus. There are no fixed due dates in the Distance Calculus online courses, so it is important that students instead set their schedules for a dedicated amount of time towards the coursework.

It is also very important to consider that going faster through a course is DIRECTLY DEPENDENT upon your math skill level, and your successful engagement of the course. We require that you complete the course in a Mastery Learning format. If you are struggling with the course content, or trying to go too fast where the quality of your submitted work is suffering, then the instructors will force a slow-down of your progress through the course, even if you have fixed deadlines.

Probability Theory Examples of the Curriculum

Below are some PDF "print outs" of a few of the Mathematica™ notebooks from Prob/Stat&Mathematica by Carpenter/Davis/Raschke/Uhl. Included as well is an example homework notebook completed by a student in the course, demonstrating how the homework notebooks become the "common blackboards" that the students and instructor both write on in their "conversation" about the notebook.

• Basics Notebook Example: 7.01.T1 - Tutorials - Monte Carlo estimation of integrals and other area measurements
• Homework Notebook Example: 7.03.G3 - Probability calculations in context: Series wiring versus parallel wiring

Distance Calculus - Student Reviews