On this page, we discuss the concept of a (discrete) Random Variable, a real-valued function defined on the sample space of a random experiment, and its associated probability distribution, probability mass function (“pmf”), and cumulative distribution function (“pdf”).

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.

  • Understand that a discrete random variable is defined by assigning values to all outcomes in the sample space of a discrete probability model.

  • Be able to give a few examples of discrete random variables.

Advanced learning objectives

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

  • Find the probability that a random variable \(X\) takes a value in any interval \(I \subseteq \mathbb{R}\) by using its probability mass function or using its cumulative distribution function.

  • Distinguish between the probability mass function and the cumulative distribution function of a discrete random variable, know the properties each must satisfy, and be able to obtain each from the other.

  • Be able to find the probability mass function of a new random variable \(Y = g(X)\), where \(X\) is a discrete random variable with a known probability mass function, and \(f: \mathbb{R} \to \mathbb{R}\).

To prepare for class

  • Watch the following videos (by Khan Academy) which introduce the concept of a random variable and its notation, and how to find its probability distribution or “probability mass function” (pmf):

  • Watch the following video (by Brendan Cordy) which introduces the notion of a probability mass function (pmf):

After class

  • Watch the following video (by Brendan Cordy) which continues the example above and introduces cumulative distribution functions (cdfs):

  • Watch the following video (by ukmathsteacher) which explains how to find the cumulative distribution function (cdf) of a random variable with several examples:

  • Watch the following video (by Khan Academy) which explains the difference between discrete and continuous random variables:

  • Watch the following video (by Katie Szeto @ MIT OpenCourseWare) which shows in detail how to find the pmf of a random variable which is a function of another random variable:


Brendan Cordy Avatar Brendan Cordy
Gabriel Indurskis Avatar Gabriel Indurskis






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