Properties of Uniform Distribution

Definition

The most basic form of continuous probability distribution function is called the uniform distribution. It is a rectangular distribution with constant probability and implies the fact that each range of values that has the same length on the distributions support has equal probability of occurrence.  This belongs to the category of maximum entropy probability distributions.

The probability density function for the variable x given that a ≤ x ≤ b is given by:

Characteristics

The following are the key characteristics of the uniform distribution:

• The density function integrates to unity
• Each of the inputs that go in to form the function have equal weighting
• Mean of the unifrom function is given by:

• The variance is given by the equation:

The plot of the uniform function is as below:

The location of the interval has little influence in deciding if the uniformly distributed variable falls within the fixed length. Two factors that influence this the most are the interval size and the fact that the interval falls within the distributions support.

• Standard uniform distribution: If a =0  and b=1 then the resulting function is called a standard unifrom distribution. This has very important practical applications.
• There are variables in physical, management and biological sciences that have the properties of a uniform distribution and hence it finds application is these fields.

Example