DEFINITION of 'Uniform Distribution'

In statistics, a type of probability distribution in which all outcomes are equally likely; each variable has the same probability that it will be the outcome. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

BREAKING DOWN 'Uniform Distribution'

There are two types of uniform distributions: discrete and continuous. The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5 or 6, but it is not possible to roll a 2.3, 4.7 or 5.5. Therefore, the roll of a die generates a discrete distribution with p = 1/6 for each outcome. A uniform distribution with only two possible outcomes is a special case of the binomial distribution.

Understanding Uniform Distributions

A distribution is a simple way to visualize a set of data, either as a graph or in a list of stating which random variables have lower or higher chances of happening. There are many different types of probability distributions, and the uniform distribution is perhaps the simplest of them all. 

Under a uniform distribution, the set of variables all have the exact same possibility of happening. This distribution, when displayed as a bar or line graph, has the same height for each potential outcome. In this way, it can look like a rectangle and therefore is sometimes described as the rectangle distribution. If you think about the possibility of drawing a particular suit from a deck of playing cards, there is a random yet equal chance of pulling a hearts as there is for pulling a spade - that is, 1/4.

Continuous Uniform Distributions

Some uniform distributions are continuous rather than discrete. An idealized random number generator would be considered a continuous uniform distribution. With this type of distribution, every variable has an equal opportunity of appearing, yet there are a continuous (or possibly infinite) number of points that can exist.

There are several other important continuous distributions, such as the normal distribution, chi-square, and Student's t-distribution.

Functions of Distributions

There are also several data generating or data analyzing functions associated with distributions to help understand the variables and their variance within a data set. These functions include probability density function, cumulative density and moment generating functions.

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