On this page, we discuss the most important discrete 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 discrete random variable which takes a finite set of values with equal probability is said to be uniformly distributed.

  • Understand the notion of a Bernoulli trial, and know that a random variable which takes the value 1 when a Bernoulli trial is successful, and 0 when unsuccessful is called a Bernoulli random variable.

  • Know that a binomial random variable counts the number of successes in a fixed number of Bernoulli trials.

  • Know that a geometric random variable counts the number of Bernoulli trials performed until the first success occurs.

  • Know that a hypergeometric random variable counts the number of successes in \(n\) trials without replacement (e.g. the number of green balls that appear in a fixed number of draws performed without replacement from an urn containing green and red balls).

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 mean and variance for uniform (on a general set \(\{a,a+1,\dots,b\}\)), Bernoulli, binomial, and geometric random variables.

  • Be able to calculate the probability that one of the above discrete random variables takes a value in a given interval.

  • Understand how and why the various discrete distributions arise in the sciences, and be able to decide which is the most appropriate in a given situation.

  • Know the definition of a Poisson random variable and be able to recognize conditions when a Poisson distribution arises in practice.

To prepare for class

  • Watch the following video (by jbstatistics) which gives a quick overview of some of the most important discrete random variables and how to recognize them:

  • After having watched the previous video, watch the following videos (all by jbstatistics) for a detailed discussion of each type of random variable and its distribution. You should at the very least focus on:

    • recognizing the precise type of situation in which the random variable applies, and
    • memorizing the formulas for the expectation and variance of the distribution.

You may also want to download and treasure this overview table of all distributions for future reference.

  • Bernoulli random variable and distribution:

  • Binomial random variable and distribution:

  • Geometric random variable and distribution:

  • Hypergeometric random variable and distribution:

After class

  • Watch the following video discussing Poisson random variables:

  • Watch the following video which shows example problems involving binomial, Poisson, hypergeometric, and geometric distributions:

  • (Optional) Watch the following videos which discuss when a random variable actually is Poisson-distributed:

  • (Optional) Watch the following video (by 3Blue1Brown) which discusses an important application of the Binomial Distribution, which will feature heavily later on in this course:


Brendan Cordy Avatar Brendan Cordy
Gabriel Indurskis Avatar Gabriel Indurskis


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