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A random variable—typically denoted by a capital letter such as X—is the numerical outcome of a random event (an "experiment"), such as: the number of heads when one flips a coin ten times; the number of accidents occurring at an intersection in a one-month period; the height of a randomly selected college basketball player. Associated with a random variable X is a probability distribution—which answers the questions "What are the possible values of X?" and "What are the probabilities associated with those values?" That is, it is a description of how the probability associated with X is spread (distributed) on the number line.
The set of possible values of a random variable is called the "support." We consider here two kinds of random variables: discrete (for which the support is either a finite or countable set), and continuous (for which the support is uncountable—usually an interval). Discrete distributions are described with a probability mass function (pmf), while continuous distributions are described with a probability density function (pdf). We typically call the pmf or pdf $f(x)$, and it must satisfy these conditions:
| Discrete (pmf) | Continuous (pdf) | |
| $f(x) \ge 0$ for all $x$ | $f(x) \ge 0$ for all $x$ | |
| $P(X=x)=f(x)$ | $P(a\le X\le b) = \int_a^b f(x)\,dx$ | |
| $\sum_{{\rm all}\ x} f(x) = 1$ | $\int_{-\infty}^\infty f(x)\,dx = 1$ | |
(There are also mixed random variables which have both discrete and continuous characteristics.)
Some of the descriptions below refer to a "sequence of Bernoulli trials." This means repeated independent attempts to achieve a "success," such as flipping a coin (trying to get heads), shooting a free throw, etc. ("Independent" means that the probability of success is constant, unaffected by previous outcomes.) When referring to Bernoulli trials, $p$ is the probability of "success," and $q=1-p$ is the probability of "failure."
In the formulas below, $\binom{n}{k}$ is the binomial coefficient "n choose k," sometimes written nCk.
Most discrete distributions have a "story" associated with them; that is, a description of a situation in which that distribution naturally arises. For example, those distributions associated with Bernoulli trials can be described in terms of a sequence of free throws or coin flips; the hypergeometric "story" might involve picking cards from a standard deck, or selecting a committee from a group of people.
Unlike discrete distributions, continuous distributions do not usually have a "story" associated with them (which is why most have little or no description given below).
(12/20/2013) Beta release: This page is far from complete, and has not been tested thoroughly. Play at your own risk. More distributions, and more features, to come.
(01/06/2014) Much more refined — but still far from perfect.
(03/12/2018) Updated (removed MooTools dependencies, fixed some bugs)
The contents of this page are © 2018 Darryl Nester. It is available to anyone who wishes to use it (like most things on the Internet). Please send me an email if you have found it to be useful, or if you have suggestions.