Uniform distribution

There are two types of uniform distribution: discrete and continuous.

The discrete case

In the discrete case, if there are N possible outcomes 1, 2, ..., N which are distributed uniformly, then the probability of outcome n is simply:

P(n) = 1 / N

A simple example of the discrete uniform distribution is throwing a fair die (or, "a dice"): Each time the die is thrown, the probability of a given score is 1/6.

The continuous case

In the continuous case, the uniform distribution is also called the rectangular distribution. It is parameterised by the smallest and largest values that the uniformly-distributed random variable can take, a and b. The probability density function of the uniform distribution is thus:

<math>

  p(x)=\left\{\begin{matrix}
  \frac{1}{b - a} & \ \ \ \mbox{for }a \leq x \leq b \\
  0 & \mbox{elsewhere}
  \end{matrix}\right.

and the cumulative distribution function is:

<math>

  F(x)=\left\{\begin{matrix}
  0 & \mbox{for }x < a \\
  \frac{x - a}{b - a} & \ \ \ \mbox{for }a \le x < b \\
  1 & \mbox{for }x \ge b
  \end{matrix}\right.

The graph of the probability density function for the continuous uniform distribution looks like:

For a random variable following this distribution, the expected value is (a + b)/2 and the standard deviation is (b - a)/√12.

The Standard Uniform distribution

The Standard Uniform Distribution is the continuous uniform distribution with the values of a and b set to 0 and 1 respectively, such that the random variable can take values between 0 and 1.

Sampling from a uniform distribution

When working with probability, it is often useful to run experiments such as computational simulations. Many programming languages have the ability to generate pseudo-random numbers which are effectively distributed according to the standard uniform distribution.

If u is a value sampled from the standard uniform distribution, then the value a + (b - a)u follows the uniform distribution parametrised by a and b, as described above. Other transformations can be used to generate other statistical distributions from the uniform distribution. In particular, the Box-Muller transformation can be used to generate samples of the normal distribution.

Uses of the uniform distribution

Although the uniform distribution is less commonly-found in nature than the normal distribution (for example), it is particularly useful for sampling from arbitrary distributions, using the inverse transform sampling method or the rejection sampling method[?].