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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 Math 315 - 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 Mathematicaâ„¢ 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:

Course: Statistics Probability Theory
Prerequisite: Algebra II from high school Calculus II (rec: Multivariable Calculus)
Intended for Majors: Humanities, Social Studies, Biological Sciences Math, Engineering, Physics, Economics, etc.


Roger Williams University Course Catalog Listing: Math 315 - Probability Theory

MATH 315: Probability Theory [3 credit hours]

Course Description: Emphasizes probability, probability density functions, distributions, statistical inferences and estimation, correlation, and regression.

Prerequisite: Calculus II (Multivariable Calculus Recommended)
Detailed Course Syllabus in PDF


Probability Theory Syllabus of Course Topics

  1. Simulations
    1. Uniform Distributions
    2. Monte Carlo Methods
    3. Random Walks
    4. Shooting craps; Iterated Fractals
  2. Data Analysis
    1. Frequency
    2. Expected Value
    3. Cumulative Distributions
    4. Variance
    5. Histograms
    6. Related formulas for Expected Values and Variance
  3. Probabilities
    1. Calculating Probability
    2. Union and Intersection and Probability
    3. Conditional Probability Formula
    4. Independence
    5. Indicator functions
  4. More Data Analysis
    1. Markov's Inequality
    2. Chebyshev's Theorem
    3. Laws of Large Numbers
    4. One-Sided Chebyshev Theorem
  5. Normal and Exponential Distributions
    1. Approximately Normally Distributed Sets
    2. Normal Distribution
    3. Approximately Exponentially Distributed Sets
    4. Exponential Distribution
    5. Memoryless Property of Exponential Distributions
  6. Random Variables
    1. Random Variables
    2. Discrete Random Variables
    3. Continuous Random Variables
    4. Probability Density Functions
    5. Cumulative Distribution Functions
    6. Expected Values and Variance
    7. Markov, Chebyshev, and Law of Large Numbers Revisited
    8. Mean, Median, and Mode
  7. Joint Distributions
    1. Joint Probability Calculations
    2. Discrete & Continuous
    3. Expected Values, Covariance, and Correlation
    4. Conditional Probability Calculations
    5. Conditional Expectations
    6. The Law of Total Probability
  8. Central Limit Theorem
    1. Generating Functions for Discrete Random Variables
    2. Generating Functions for Continuous Random Variables
    3. Generating Functions and Independence
    4. Central Limit Theorem
    5. Chi-squared and Gamma random variables
  9. Counting
    1. Binomial and Poisson counting
    2. Binomial and Poisson Distributions
    3. Sampling
    4. Confidence Intervals
    5. Hypothesis testing


Probability Theory Examples of the Curriculum

Below are some PDF "print outs" of a few of the Mathematicaâ„¢ 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.








Distance Calculus - Student Reviews

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Date Posted: Sep 20, 2020
Review by: Genevieve P.
Courses Completed: Applied Calculus
Review: I found out from my grad school after being accepted that I needed a Calculus course before starting their MBA program. I had less than 6 weeks to do it (and as a non-STEM undergrad no less). The video lectures were informative, the pre-calc refresher was great to get re-conditioned, and the asynchronous format worked so well as I did this at night/weekends after work. I completed it in 4 weeks. Professor Curtis was extremely responsive, graded assignments quickly, and a supportive guide providing constructive feedback to me to excel at the assignments. I highly recommend this course for those who need a pre-req in a hurry or like learning on their own schedule. Thanks, Distance Calculus and Professor Curtis!
Transferred Credits to: Massachusetts Institute of Technology (MIT)





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Date Posted: Jun 6, 2020
Review by: Douglas Z.
Courses Completed: Multivariable Calculus, Differential Equations, Linear Algebra, Probability Theory
Review: I loved these courses. So in depth and comprehensive. The mix of software and math curriculum was tremendously helpful to my future studies and career in engineering. I highly recommend these courses if you are bored of textbook courses.
Transferred Credits to: University of Massachusetts, Amherst





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Date Posted: Dec 8, 2020
Review by: Aileen C.
Courses Completed: Differential Equations
Review: This course may be more difficult than your average differential equations course, which better prepares you to use these skills in your degree. The self-learning does make learning some of the concepts challenging, but you get the help you need to understand these concepts.
Transferred Credits to: Johns Hopkins University





Frequently Asked Questions

No. The actual topic coverage of Statistics and Probability are very close to one another. The Probability Theory course does everything with the machinery of Calculus, while the Statistics course stays away from Calculus and just concentrates on observing the patterns in the data.

Yes. Probability Theory is exactly a first course in calculus-based Statistics.

Yes. It is mandatory that any Data Science student will have taken Probability Theory (and more advanced courses after Probability Theory as well) prior to starting a Data Science certificate or degree.

No. Calculus II is very much a prerequisite for Probability Theory, and not a nominal prerequisite. Probability Theory relies very strongly on the mastery of the Calculus II content.

Yes, All Distance Calculus courses are offered through Roger Williams University in Providence, Rhode Island, USA, which is regionally accredited (the highest accreditation) through New England Commission of Higher Education (NECHE).