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Brilliant

Random Variables & Distributions

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Overview

Random variables and their distributions are the best tools we have for quantifying and understanding unpredictability. This course covers their essential concepts as well as a range of topics aimed to help you master the fundamental mathematics of chance.

Upon completing this course, you'll have the means to extract useful information from the randomness pervading the world around us.

Syllabus

  • Introduction: What makes a variable random?
    • Random Variables: How do you quantify chance in a dicey world?
    • Distributions: Get a glimpse of how probabilities for random variables are computed.
    • Random Variable Applications: Explore some of the many real-world uses of random variables.
  • Discrete Random Variables: The language of random variables: independence, distributions, and more.
    • Definition: What is a discrete random variable?
    • Density Functions: Learn the process for assigning probabilities to random variables.
    • Joint Distributions: Tackle problems with two uncertain quantities.
  • Expected Value: Know what outcome to expect when you're dealing with randomness.
    • Expected Value Definition and Properties: Use averages to make predictions about random events.
    • Expected Value Calculations: Gain hands-on experience with expectation value by exploring real-world applications.
    • Conditional Expectation: Practice refining your expectations based on new information.
    • Linearity of Expectation: Explore the most important feature of the expected value.
    • Indicator Variables: Unleash the power of linearity on challenging probability problems.
  • Variance: It's the mathematical way to describe how erratic your random variable is.
    • Variance Definition and Properties: Develop an important means for assessing expected values.
    • Variance and Standard Deviation: Calculate the spread of possibilities for a range of real-world scenarios.
    • Covariance: Learn how to measure how two variables influence each other.
  • Discrete Distributions: Use these models to connect the theory to the real-world.
    • Uniform Discrete Distribution: In this type of distribution, all outcomes are equally likely.
    • Bernoulli Distribution: Investigate what surveys and coin flips have in common.
    • Binomial Distribution: Apply the technique of Bernoulli trials to challenging probability problems.
    • Geometric Distribution: Discover what it means for a distribution to have "no memory."
    • Poisson Distribution: Explore one of the most versatile and widely-used distributions.
    • Applications of Discrete Distributions: Practice what you've learned with sports applications and games of chance.
  • Continuous Random Variables: When the world gets continuous, calculus meets probability.
    • Definition: Understand what it means for a random variable to have uncountably many outcomes.
    • Density Functions: Extend the rules of probability to the infinite.
    • Joint Distributions: Make sense of the uncertainty in two continuous random variables.
    • Expected Value and Variance: Calculate predictions for uncountable outcomes and learn how to judge their accuracy.
  • Continuous Distributions: Model heights, stocks, or just about anything else with these distributions.
    • Normal Distribution: Learn about a type of continuous distribution that you see pop up everywhere!
    • Exponential Distribution: Discover why there's a common distribution for radioactivity and for winning the lottery.
    • Gamma Distribution: Explore a distribution used by insurance companies the world over.
    • Log-Normal Distribution: Learn to model stock market fluctuations in a simple way.

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