Continuous Distribution Zoo

Overview

On this page, we discuss the most important continuous random variables and their distributions.

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

• Know that a continuous random variable whose pdf is constant on some interval and zero everywhere else is said to be uniformly distributed on that interval. Be able to recognize whether a continuous random variable is uniformly distributed from its pdf.

• Know the definition of an exponentially distributed continuous random variable, and be able to recognize whether a continuous random variable is exponentially distributed from its pdf.

• Know the definition of a Pareto distributed continuous random variable, and be able to recognize whether a continuous random variable is Pareto distributed from its pdf.

• Know the definition of a Gaussian distribution (also known as a Normal distribution) and be able to standardize the values of a Gaussian random variable by computing z-scores.

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

• Be able to derive and use the formulas for the expected value and variance of a uniformly distributed continuous random variable.

• Be able to derive and use the formulas for the expected value and variance of an exponentially distributed random variable.

• Understand the memoryless property of exponentially distributed random variables.

• Be able to derive and use the formulas for the expected value and variance of a Pareto distributed random variable.

• Understand that the expected value and variance of a Pareto distributed random variable may be undefined, and the more general notion of a fat-tailed distribution.

• Use a z-table to find the probability that a continuous random variable with the standard normal distribution takes a value in any interval.

• Combine this procedure with the idea of standardization to find the probability that a random variable with any Gaussian distribution takes a value in any interval.

• Use the 68-95-99.7 Rule of Thumb to quickly estimate probabilities for normally distributed random variables.

To prepare for class

• Watch the following video (by jbstatistics) which gives an introduction to the continuous uniform distribution:

• Watch the following video (by MIT OpenCourseWare) which introduces the exponential distribution and explains its relationship to the discrete geometric distribution:

• Watch the following video (by MindYourDecisions) which introduces the Pareto principle underlying the Pareto distribution:

• Watch the following video (by jbstatistics) which gives an introduction to the normal or Gaussian distribution:

• Watch the following video (by Brendan Cordy) which shows that the mean of a normal distributed random variable with parameter \(\mu=0\) is indeed \(0\), and shows an example of how to standardize a non-standard normal distribution:

• Play around with the following interactive web apps:

  • Interactive z-table app (by Brendan Cordy) to visualize & calculate probabilities for a standard normal distribution.

  • Normal distribution app (by Matt Bognar) to visualize & calculate probabilities for different (non-standard) normal distributions (by changing the parameters \(\mu\) and \(\sigma\)).

  • Web apps for various continuous distributions (by Matt Bognar) (there is also a free app for smartphones) to visualize & calculate probabilities for many (discrete and) continuous distributions (in particular exponential, Pareto, and normal distributions).

• Use this cumulative standard normal distribution table to calculate some probabilities for normal distributions by hand. Note that such z-tables are nowadays a bit of a mathematical artifact, since we can now easily use a computer or smartphone app instead - but studying these tables can be nevertheless useful, and is often still used on various standardized tests.

After class

• Watch the following video (by MIT OpenCourseWare) which explains the memorylessness property of the exponential distribution, and how it can be seen as the counterpart to the discrete geometric distribution:

• Watch the following video (by jbstatistics) which explains how to standardize a normal distribution:

• Watch the following video (by Brendan Cordy) which shows example calculations with various normal distributed random variables:

• Watch the following video (by Joshua Emmanuel) which explains the 68-95-99.7% rule for normal distributions and shows how to use it to quickly estimate various probabilities:

Optional: For further enrichment

• Watch the following video (by jbstastics) which discusses how (& when) to use the Normal distribution to approximate a Binomial distribution - and how to use a continuity correction to make this approximation better:

• Watch the following sequence of videos (by MIT OpenCourseWare) which introduce the Poisson process and Poisson distribution, and show applications and example calculations involving these:

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