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Uniform distribution

The uniform distributions are simple probability distributions which, in the discrete case, can be characterized by saying that all possible values are equally probable.

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

Table of contents

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.
</math>

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.
</math>

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

The continuous uniform probability distribution

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[?].



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