Foundations of Probability and Random Variables
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Foundations of Probability and Random Variables
This course is part of Statistical Methods for Computer Science Specialization
Instructors: Ian McCulloh
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What you'll learn
Master combinatorial techniques, including permutations, combinations, and multinomial coefficients, to solve counting and probability problems.
Apply probability axioms, construct Venn diagrams, and calculate sample space sizes to evaluate probabilities in various scenarios.
Utilize Bayes' formula, the multiplication rule, and conditional probability to assess event relationships and solve real-world problems.
Analyze discrete and continuous random variables using probability density functions, cumulative distribution functions, and expected values.
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There are 6 modules in this course
The course "Foundations of Probability and Random Variables" introduces fundamental concepts in probability and random variables, essential for understanding computational methods in computer science and data science. Through five comprehensive modules, learners will explore combinatorial analysis, probability, conditional probability, and both discrete and continuous random variables. By mastering these topics, students will gain the ability to solve complex problems involving uncertainty, design probabilistic models, and apply these concepts in fields like machine learning, AI, and algorithm design.
What makes this course unique is its practical approach: students will develop hands-on proficiency in the R programming language, which is widely used in data science and statistical modeling. The course also includes real-world applications, allowing learners to bridge theoretical knowledge with practical problem-solving skills. Whether you are aiming to pursue advanced studies in machine learning or develop data-driven solutions in professional settings, this course provides the solid foundation you need to excel. Designed for learners with a background in calculus and basic programming, this course prepares you to tackle more advanced topics in computational science.
This course provides a comprehensive introduction to fundamental concepts in probability and statistics, focusing on counting principles, permutations, combinations, and multinomial coefficients. You will explore probability axioms, conditional probabilities, and Bayes’s Formula while using Venn diagrams to visualize events. The course covers random variables, including discrete and continuous types, expected values, and various probability distributions. Practical applications in R programming and data analysis tools will enhance understanding through simulations and real-world problem-solving. By the end, you will be equipped to analyze and interpret statistical data effectively.
What's included
2 readings1 plugin
2 readings•Total 10 minutes
- Course Overview•5 minutes
- Instructor Biography - Dr. Tony Johnson•5 minutes
1 plugin•Total 4 minutes
- Instructor Biography - Dr. Ian McCulloh•4 minutes
This module covers the usefulness of an effective method for counting the number of ways that things can occur. Many problems in probability theory can be solved simply by counting the number of different ways that a certain event can occur.
What's included
9 videos2 readings3 assignments1 ungraded lab
9 videos•Total 116 minutes
- Introduction to Data Science•20 minutes
- Basic Principle of Counting•7 minutes
- Generalized Principle of Counting•4 minutes
- Permutations and Combinations•18 minutes
- Circular Permutations•10 minutes
- Combinations•19 minutes
- Factorials and Identity•18 minutes
- Distributing Indistinguishable Items•8 minutes
- R Tutorial•13 minutes
2 readings•Total 90 minutes
- Reading References•45 minutes
- Reading References•45 minutes
3 assignments•Total 90 minutes
- Combinatorial Analysis•60 minutes
- Fundamentals of Counting in Data Science•15 minutes
- Mastering Combinatorial Techniques: From Combinations to R Applications•15 minutes
1 ungraded lab•Total 60 minutes
- Practice Lab: Exploring Combinatorics and Permutations Using R•60 minutes
This module introduces the concept of the probability of an event and then shows how probabilities can be computed in certain situations.
What's included
9 videos3 readings4 assignments1 ungraded lab
9 videos•Total 141 minutes
- Sample Spaces•14 minutes
- Events•12 minutes
- Venn Diagram•10 minutes
- DeMorgan Laws•11 minutes
- Axioms of Probability•16 minutes
- Simple Propositions•22 minutes
- Equally Likely Outcomes•15 minutes
- ELO Example•13 minutes
- R Tutorial•29 minutes
3 readings•Total 180 minutes
- Reading References•60 minutes
- Reading References•60 minutes
- Reading References•60 minutes
4 assignments•Total 105 minutes
- Probability•60 minutes
- Understanding Probability: Sample Spaces, Events, and Venn Diagrams•15 minutes
- Foundations of Probability: DeMorgan's Laws and Axioms•15 minutes
- Exploring Probability: Simple Propositions, Equally Likely Outcomes, and R Tutorial•15 minutes
1 ungraded lab•Total 60 minutes
- Practice Lab: Understanding Probability and Combinatorics Using R•60 minutes
This module explores one of the most important concepts in probability theory, that of conditional probability. The importance of this concept is twofold. First, you will be interested in calculating probabilities when some partial information concerning the result of an experiment is available; in such a situation, the desired probabilities are conditional. Second, even when no partial information is available, conditional probabilities can often be used to compute the desired probabilities more easily.
What's included
8 videos3 readings4 assignments1 ungraded lab
8 videos•Total 73 minutes
- Conditional Probability•10 minutes
- Example Cond Prob•14 minutes
- Reb Balls from Urn•5 minutes
- Revisit Bayes Rule•11 minutes
- Independence•7 minutes
- Ex Medical Testing•12 minutes
- Paradox of the Carnival Dice•9 minutes
- Paradox of the Discrimination Lawsuit•6 minutes
3 readings•Total 360 minutes
- Reading References•120 minutes
- Reading References•120 minutes
- Reading References•120 minutes
4 assignments•Total 105 minutes
- Conditional Probability and Independence•60 minutes
- Conditional Probability and Practical Examples•15 minutes
- Bayes' Rule and Probability Independence•15 minutes
- Exploring Probability Paradoxes and Real-World Applications•15 minutes
1 ungraded lab•Total 60 minutes
- Practice Lab: COVID-19 Probability Models and Testing Scenarios in R •60 minutes
This module discusses the function of outcomes rather than the actual outcomes themselves. In particular, you will examine random variables that can take on at most a countable number of possible values. You can call these types of variables, discrete random variables.
What's included
9 videos4 readings5 assignments1 ungraded lab
9 videos•Total 176 minutes
- Random Variables•16 minutes
- R.V. Coin Toss•15 minutes
- Coin Toss Proof•9 minutes
- Expected Value•19 minutes
- Expectation of R.V. Function•15 minutes
- Variance of R.V.•13 minutes
- Bernoulli R.V. and Mass Functions•43 minutes
- Defective Product Example•19 minutes
- R Tutorial•28 minutes
4 readings•Total 480 minutes
- Reading References•120 minutes
- Reading References•120 minutes
- Reading References•120 minutes
- Reading References•120 minutes
5 assignments•Total 120 minutes
- Discrete Random Variables•60 minutes
- Introduction to Random Variables and Coin Tosses•15 minutes
- Understanding Expected Value and Random Variables•15 minutes
- Variance and Bernoulli Random Variables•15 minutes
- Analyzing Defective Products and R Tutorial•15 minutes
1 ungraded lab•Total 60 minutes
- Practice Lab: Statistical Computation and Simulation Using R•60 minutes
This module extends the concept of random variables where the outcomes cannot be counted. You will explore probability density functions, cumulative distribution functions, the normal distribution and other common distributions.
What's included
10 videos4 readings5 assignments1 ungraded lab
10 videos•Total 118 minutes
- Continuous RV•10 minutes
- PDF and CDF•13 minutes
- PDF and CDF Example•5 minutes
- Means and Expectation•8 minutes
- Uniform PDF Example•11 minutes
- Cumulative Distribution Function (CDF)•11 minutes
- The Normal Distribution•17 minutes
- Normal Distribution Example•8 minutes
- Other Distributions and the Hazard Rate•20 minutes
- R Tutorial•17 minutes
4 readings•Total 360 minutes
- Reading References•90 minutes
- Reading References•90 minutes
- Reading References•90 minutes
- Reading References•90 minutes
5 assignments•Total 120 minutes
- Continuous Random Variables•60 minutes
- Continuous Random Variables: PDF and CDF Basics•15 minutes
- Means, Expectation, and Uniform PDF Example•15 minutes
- Understanding CDF and the Normal Distribution•15 minutes
- Exploring Distributions, Hazard Rates, and R•15 minutes
1 ungraded lab•Total 60 minutes
- Practice Lab: Statistical Simulations and Probability Modeling in R•60 minutes
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