Probability and Statistics II: Random Variables – Great Expectations to Bell Curves from edX

This course discusses properties and applications of random variables. When you’re done, you’ll have enough firepower to undertake a wide variety of modeling and analysis problems; and you’ll be well-prepared for the upcoming Statistics courses.

We’ll begin by introducing the concepts of discrete and continuous random variables. For instance, how many customers are likely to arrive in the next hour (discrete)? What’s the probability that a lightbulb will last more than a year (continuous)?

We’ll learn about various properties of random variables such as the expected value, variance, and moment generating function. This will lead us to a discussion of functions of random variables. Such functions have many uses, including some wonderful applications in computer simulations.

If you enjoy random variables, then you’ll really love joint (two-dimensional) random variables. We’ll provide methodology to extract marginal (one-dimensional) and conditional information from these big boys. This work will enable us to study the important concepts of independence and correlation.

Along the way, we’ll start working with the R statistical package to do some of our calculations and analysis.

What's inside

Learning objectives

Upon completion of this course, learners will be able to:

• identify discrete and continuous random variables• describe the properties of random variables, including the expected value, variance, and moment generating function.• understand functions of random variables, and how they can be used in computer simulation applications.• recognize joint (two-dimensional) random variables and how to extract marginal (onedimensional) and conditional information from them• implement the concepts of independence and correlation• work with and implement the r statistical package

Upon completion of this course, learners will be able to:
• identify discrete and continuous random variables• describe the properties of random variables, including the expected value, variance, and moment generating function.• understand functions of random variables, and how they can be used in computer simulation applications.• recognize joint (two-dimensional) random variables and how to extract marginal (onedimensional) and conditional information from them• implement the concepts of independence and correlation• work with and implement the r statistical package

Syllabus

“FCPS” refers to the free text, A First Course in Probability and Statistics: free access is provided via a PDF file or as a book

Module 1: Univariate Random Variables

Lesson 1: Introduction (FCPS §2.1)

Lesson 2: Discrete Random Variables (FCPS §2.2)

Lesson 3: Continuous Random Variables (FCPS §2.3)

Lesson 4: Cumulative Distribution Functions (FCPS §2.4)

Lesson 5: Great Expectations (FCPS §2.5.1)

Lesson 6: LOTUS, Moments, and Varience (FCPS §2.5.2)

Lesson 7 [OPTIONAL]: Approximations to E[h(X)] and Var(h(X)) (FCPS §2.5.3)

Lesson 8: Moment Generating Functions (FCPS §2.6)

Lesson 9: Some Probability Inequalities (FCPS §2.7)

Lesson 10: Functions of a Random Variable (FCPS §2.8.1)

Lesson 11: Inverse Transform Theorem (FCPS §2.8.2)

Lesson 12 [OPTIONAL]: Honors Bonus Results (FCPS §2.8.3)

Module 2: Bivariate Random Variables

Lesson 1: Introduction (FCPS §§3.1.1−3.1.3)

Lesson 2: Marginal Distributions (FCPS §3.1.4)

Lesson 3: Conditional Distributions (FCPS §3.2)

Lesson 4: Independent Random Variables (FCPS §3.3.1)

Lesson 5: Consequences of Independence (FCPS §3.3.2)

Lesson 6: Random Samples (FCPS §3.3.3)

Lesson 7: Conditional Expectation (FCPS §3.4.1)

Lesson 8: Double Expectation (FCPS §3.4.2)

Lesson 9 [OPTIONAL]: First-Step Analysis (FCPS §3.4.3)

Lesson 10 [OPTIONAL]: Random Sums of Random Variables (FCPS §3.4.3)

Lesson 11 [OPTIONAL]: Standard Conditioning Argument (FCPS §3.4.3)

Lesson 12: Covariance and Correlation (FCPS §3.5.1)

Lesson 13: Correlation and Causation (FCPS §3.5.2)

Lesson 14: A Couple of Worked Correlation Examples (FCPS §3.5.3)

Lesson 15: Some Useful Covariance / Correlation Theorems (FCPS §3.5.4)

Lesson 16: Moment Generating Functions, Revisited (FCPS §3.6)

Lesson 17 [OPTIONAL]: Honors Bivariate Functions of Random Variables (FCPS §3.7)

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Develops understanding of probability and random variables, which are vital knowledge in statistics and many disciplines

Utilizes the R statistical package, equipping learners with practical skills in statistical analysis

Provides a strong foundation for further studies in statistics or related fields

Appropriate for learners with some background in probability and calculus

Covers marginal and conditional distributions, key concepts in probability theory

Taught by David Goldsman, recognized for his expertise in probability and statistics

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Probability and Statistics II: Random Variables – Great Expectations to Bell Curves with these activities:

Review course syllabus and go through the course outline

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Refreshes the prior knowledge by going over syllabus in order to get an overview of the topics covered in the course.

Browse courses on Random Variables

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Access the course outline on the course website
Review the topics that will be covered in each module
Identify any topics that you may need to review
Note down any questions or concerns that you may have

Read a book on statistical distributions

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Reading a book on statistical distributions will provide you with a comprehensive understanding of the different types of distributions and their applications.

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Purchase or borrow a copy of the book.
Read the book and take notes.
Solve the practice problems at the end of each chapter.

Review the concept of random variables

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Reviewing the concept of random variables will provide a strong foundation for this course and enhance your understanding of subsequent topics.

Browse courses on Random Variables

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Define a random variable and identify its characteristics.
Classify random variables as discrete or continuous.
Calculate the probability mass function for discrete random variables.
Calculate the probability density function for continuous random variables.

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Solve practice problems on random variables

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Solving practice problems will reinforce your understanding and build your problem-solving skills.

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Find practice problems in a textbook or online.
Solve the problems and check your answers.
Review your mistakes and learn from them.

Seek out a mentor or tutor

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Connecting with a mentor or tutor can provide you with additional support and guidance throughout the course.

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Identify potential mentors or tutors in your network or through online platforms.
Reach out and schedule a meeting to discuss your goals and needs.
Meet with your mentor or tutor regularly to discuss the course material and ask questions.

Follow online tutorials on joint random variables

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Following online tutorials will provide you with additional insights and examples on joint random variables and their properties.

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Search for online tutorials on joint random variables.
Watch the tutorials and take notes.
Practice the examples and exercises provided in the tutorials.

Create a visual representation of probability distributions

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Creating visual representations of probability distributions will help you develop a deeper understanding of their shapes and properties.

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Select a random variable and collect data.
Create a histogram or scatterplot to represent the data.
Analyze the shape and properties of the distribution.

Participate in a data analysis or modeling competition

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Participating in a competition will challenge you to apply your knowledge and skills in a practical setting.

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Find a data analysis or modeling competition that aligns with your interests.
Register for the competition and form a team or work independently.
Analyze the data and develop a model or solution.
Submit your solution to the competition.

Career center

Learners who complete Probability and Statistics II: Random Variables – Great Expectations to Bell Curves will develop knowledge and skills that may be useful to these careers:

Data Analyst

Data Analysts take raw data and turn it into valuable information that businesses can use to make better decisions. They use statistical techniques to analyze data, identify trends, and develop predictive models. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Data Analysts use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis. It also introduces students to the R statistical package, which is widely used by Data Analysts.

See salaries and explore the career path for Data Analyst

Statistician

Statisticians use statistical techniques to collect, analyze, and interpret data. They work in a variety of fields, including healthcare, finance, and marketing. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Statisticians use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Financial Analyst

Financial Analysts use statistical techniques to analyze financial data and make investment recommendations. They use probability distributions to model the risk and return of different investments. They also use regression analysis to identify trends in financial data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Financial Analysts use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Risk Analyst

Risk Analysts use statistical techniques to assess the risk of different events. They use probability distributions to model the likelihood of different events occurring. They also use regression analysis to identify trends in risk data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Risk Analysts use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Market Researcher

Market Researchers use statistical techniques to collect and analyze data about consumer behavior. They use probability distributions to model the behavior of consumers. They also use regression analysis to identify trends in consumer behavior. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Market Researchers use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Quantitative Analyst

Quantitative Analysts use statistical techniques to develop and implement trading strategies. They use probability distributions to model the risk and return of different investments. They also use regression analysis to identify trends in financial data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Quantitative Analysts use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Biostatistician

Biostatisticians use statistical techniques to design and analyze clinical trials. They use probability distributions to model the likelihood of different outcomes occurring. They also use regression analysis to identify trends in clinical data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Biostatisticians use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Underwriter

Underwriters use statistical techniques to assess the risk of different insurance policies. They use probability distributions to model the likelihood of different events occurring. They also use regression analysis to identify trends in insurance data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Underwriters use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Machine Learning Engineer

Machine Learning Engineers use statistical techniques to develop and implement machine learning algorithms. They use probability distributions to model the data that the algorithms will be trained on. They also use regression analysis to identify trends in the data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Machine Learning Engineers use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Data Journalist

Data Journalists use statistical techniques to analyze data and tell stories. They use probability distributions to model the likelihood of different events occurring. They also use regression analysis to identify trends in data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Data Journalists use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Operations Research Analyst

Operations Research Analysts use statistical techniques to solve business problems. They use probability distributions to model the risk and return of different decisions. They also use regression analysis to identify trends in business data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Operations Research Analysts use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Software Engineer

Software Engineers use statistical techniques to develop and test software. They use probability distributions to model the likelihood of different errors occurring. They also use regression analysis to identify trends in software performance. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Software Engineers use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Product Manager

Product Managers use statistical techniques to analyze customer data and make decisions about product development. They use probability distributions to model the likelihood of different customers buying a product. They also use regression analysis to identify trends in customer behavior. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Product Managers use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Actuary

Actuaries use statistical techniques to assess the risk of different financial products and services. They use probability distributions to model the likelihood of different events occurring. They also use regression analysis to identify trends in financial data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Actuaries use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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Epidemiologist

Epidemiologists use statistical techniques to study the causes and spread of diseases. They use probability distributions to model the likelihood of different diseases occurring. They also use regression analysis to identify trends in disease data. The Probability and Statistics II course provides a strong foundation in the statistical techniques that Epidemiologists use every day. The course covers topics such as probability distributions, hypothesis testing, and regression analysis.

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