Uniform Distribution - Continuous

The uniform distribution describes a variable where the probability of occurrence of any value between in range defined by the minimum and maximum values is equal.

An important application of the uniform distribution is a form of random number generator which uses a cumulative probability curve to transform a uniform variable to a non-random one.

A continuous distribution with minimum=0 and maximum=1 is often referred to as a standard uniform distribution.

Profile

Parameters

Parameter	Description	Characteristics
min	Minimum value	A float > -∞ and < ∞
max	Maximum value	A float > -∞ and < ∞ and > min

Functions

Properties

The profile of the P(x) does not change with the parameters, thus skewness and kurtosis are fixed values.

Example

Examples of continuous uniform distribution include composition samples from perfect mixtures, intervals within certain atomic processes and the number of seconds from the start of an hour of requests on we sever as shown in the graph below.

Whilst it might be possible to estimated the number of requests per hour to a web server, it is not possible to estimate when those requests will arrive.

Random Number Generation

Random number generation (referred to as R) for a continuous uniform distribution can be performed by transforming a standard continuous uniform variable (referred to as U) with the inverse probability function:

r=g(u)

Using Basic style code, the function would be similar to:

r=min+rnd*(max-min)

Parameter Estimation

The matching moment equations for the minimum and maximum parameters are obtained by rearranging the equations for the mean and variance:>