What Is a Uniform Distribution?

Graph of the probability density function for a uniform distribution
Graph of the probability density function for a uniform distribution. C.K.Taylor

There are a number of different probability distributions. Each of these distributions has a specific application and use that is appropriate to a particular setting. These distributions range from the ever-familiar bell curve (aka a normal distribution) to lesser known such as the gamma distribution. Most distributions involve a complicated density curve, but there are some that do not. One of the simplest density curves is for a uniform probability distribution.

Features of the Uniform Distribution

The uniform distribution gets its name from the fact that the probabilities for all outcomes are the same. Unlike a normal distribution with a hump in the middle, or a chi-square distribution, a uniform distribution has no mode. Instead every outcome is equally likely to occur. Unlike a chi-square distribution, there is no skewness to a uniform distribution. As a result, the mean and median coincide.

Since every outcome in a uniform distribution occurs with the same relative frequency, the resulting shape of the distribution is that of a rectangle.

Uniform Distribution for Discrete Random Variables

Any situation in which every outcome in a sample space is equally likely will use a uniform distribution. One example of this in a discrete case is when we roll a single standard die. There are a total of six sides of the die, and each side has the same probability of being rolled face up.

The probability histogram for this distribution is rectangular shaped, with six bars that each have height of 1/6.

Uniform Distribution for Continuous Random Variables

For an example of a uniform distribution in a continuous setting, we will consider an idealized random number generator. This will truly generate a random number from a specified range of values.

So if we specify that the generator is to produce a random number between 1 and 4, then 3.25, 3, e, 2.222222, 3.4545456 and pi are all possible numbers that are equally likely to be produced.

Since the total area enclosed by a density curve must be 1, which corresponds to 100%, it is straightforward to determine the density curve for our random number generator. If the number is from the range a to b, then this corresponds to an interval of length b - a. In order to have an area of one, the height would have to be 1/(b - a) .

For an example of this, for a random number generated from 1 to 4, the height of the density curve would be 1/3.

Probabilities with a Uniform Density Curve

It is important to remember that the height of a curve does not directly indicate the probability of an outcome. Rather, as with any density curve, probabilities are determined by the areas under the curve.

Since a uniform distribution is shaped like a rectangle, the probabilities are very easy to determine. Rather than using calculus to find the area under a curve, we can simply use some basic geometry. All that we need to remember is that the area of a rectangle is its base multiplied by its height.

We will see this by returning to the same example that we have been studying.

In this illustration we saw that Xis a random number generated between the values 1 and 4, the probability that X is between 1 and 3 is 2/3, because this constitutes the area under the curve between 1 and 3.