Uniform distribution may refer to discrete uniform distribution or continuous uniform distribution. It is a distribution that has constant probability and is known as a rectangular distribution. The distribution is abbreviated as U(a, b).
When the distribution is equally spaced and when the probability density is same at any point, the uniform distribution is called â€˜discrete uniform distributionâ€™. The other type of continuous distribution is a case where the variable can be assigned with infinite number of values known as continuous uniform distribution.
Examples : Generation of random numbers, time of occurrence in a particular interval etc.,

## Properties

Listed below are few properties of uniform distribution:
1) Let a and b are the two parameters of the distribution. The distribution is called uniform distribution on (a, b) since it assumes a constant value for all x in (a, b).

2) The distribution is also known as rectangular distribution, since the curve y = f(x) describes a rectangle over the x axis and between the ordinates at x = a and x = b.

3) A uniform or rectangular variate X on the interval (a, b) is written as :

X $\sim$ U[a, b]  or X $\sim$ R [a, b]

4) The cumulative distribution function $F(x)$ is given by:

$F(x)$ = $\left\{\begin{matrix} 0, x \leq a\\ \frac{x-a}{b-a}, a < x < b\\ 1, x \geq b \end{matrix}\right.$

Since $F(x)$ is not continuous at x = a and x = b, it is not differentiable at these points.

$\frac{d}{dx}$ $F(x)$ = $f(x)$ = $\frac{1}{b - a }$ $\neq$ 0, exists everywhere except at the points x = a and x = b and the pdf $f(x)$ is

$f(x ; a, b)$= $\left\{\begin{matrix} \frac{1}{b - a }, if\ a < x < b\\ 0, otherwise\end{matrix}\right.$

5) For a uniform variate X in (-a, a) the pdf is given by
$f(x)$ = $\left\{\begin{matrix} \frac{1}{2a}, -a < x < a\\ 0, otherwise\end{matrix}\right.$

## Mean and Variance

The formula for mean and variance of continuous uniform distribution are described below:
Mean value of this distribution is $\frac{(a + b)}{2}$

The variance of this distribution is $\frac{(b - a)^{2}}{12}$
Where a and b are the two parameters of the distribution.

## Probability Distributions

Probability density function of uniform distribution  is defined as :

$f(x)$ = $\left\{\begin{matrix} \frac{1}{b-a} & a<x<b \\ 0 & otherwise \end{matrix}\right.$

a and b are the parameters of the uniform distribution.

If a random variable X admits a uniform density function, then X is said to be uniformly distributed or X has the uniform distribution.
A probability density function is a function f defined on an interval (a, b) and should have the following properties.
a) $f(x)$  $\geq$ 0 for every x

b) $\int_{a}^{b}$ f(x)dx = 1

## Discrete Uniform Distribution

A discrete uniform is the simplest type of all the probability distributions.
Discrete uniform distribution is known as the equally likely outcomes distribution and the shorthand X $\sim$ discrete U(a, b) is used to indicate that the random variable X has the discrete uniform distribution with integer parameters a and b, where a < b. A discrete uniform random variable X with parameters a and b has probability mass function $f(x)$ = $\frac{1}{b - a + 1}$     for x = a, a + 1, ............., b

Here each of the n values in its range say x$_{1}$, x$_{2}$,........, x$_{n}$ has equal probability.

Then f(x$_{i}$) = $\frac{1}{n}$

Here n is known as the parameter of the distribution and lies in the set of all positive integers.
f(x) represents the probability mass function. This distribution is appropriate for a die experiment and for an experiment with a deck of cards.

Example : Suppose we throw a die. Let X be the random variable denoting what number is thrown.

Then we have output like, P(X = 1) = $\frac{1}{6}$, P(X = 2) = $\frac{1}{6}$ etc.

In fact, P(X = x) = $\frac{1}{6}$  for all x between 1 and 6. Hence we have a uniform distribution.

## Continuous Uniform Distribution

The continuous uniform distribution is the probability distribution of random numbers from the continuous interval between a and b.
Its density function is defined by the following.
When a $\leq$ x $\leq$ b

=> $f(x)$ = $\frac{1}{b - a}$

when x < a or x > b

=> f(x) = 0
If a continuous random variable X can assume any value in the interval a $\leq$ x $\leq$ b and only these values, and if its probability density function f(x) is constant over that interval and equal to zero, then X is said to be uniformely distributed and its distribution is called a continuous uniform probability distribution.

The pdf of continuous uniform distribution is shown in the graph below:

## Examples

Examples of uniform distribution is given below :
Example 1 : If X is uniformly distributed with mean 1 and variance $\frac{4}{3}$, find P(X < 0)

Solution : Let X $\sim$ U[a, b] so that

$p(x)$ = $\frac{1}{b - a}$, a < x < b

Given : Mean = $\frac{1}{2}$(b + a) = 1
b + a = 2

Var (X) = $\frac{1}{12}$ (b - a)$^{2}$ = $\frac{4}{3}$

b - a = $\pm$ 4

On solving we get a = -1 and b = 3 => (a < b)

p(x) = $\frac{1}{4}$; -1 < x < 3

P(X < 0) =  $\int_{-1}^{0}$ p(x) dx

= $\frac{1}{4}$ |x|$_{-1}^{0}$

= $\frac{1}{4}$

Example 2 : If X is a random variable with a continuous distribution function F, then prove that F(X) has a uniform distribution on [0, 1].
Prove that P[ a $\leq$ F(x) $\leq$ b] = b - a, 0 $\leq$ (a, b) $\leq$ 1

Solution : Since F is a distribution function, it is non-decreasing. Let Y = F(X)

Then the distribution function G of Y is given by:

G$_{Y}$(y) = P(Y $\leq$ y)

= P [F(X) $\leq$ y]

= P [ X $\leq$  F$^{-1}$(y)]

Inverse of function exists since F is non decreasing and given to be continuous.

G$_{Y}$(y) = F[F$^{-1}$(y)]

since F is the distribution function of X.

Thus G$_{Y}$(y) = y

Therefore the pdf of Y = F(X) is given by ;

g$_{Y}$(y) = $\frac{d}{dy}$[G$_{Y}$(y)] =  1

Since F is a distribution function Y = F(X) takes the value in the range [0, 1].

Hence g$_{Y}$(y) = 1, 0 $\leq$ y $\leq$ 1

Y is a uniform variate on [0, 1].

Since Y = F(X)  $\sim$ U[0, 1]

P[ a $\leq$ F(X) $\leq$ b] = P(a $\leq$ Y $\leq$ b]

= $\int_{a}^{b}$ g(y) dy

= $\int_{a}^{b}$ 1. dy

= |y|$_{a}^{b}$

= b - a