14.6 - Uniform Distributions

Uniform Distribution

A continuous random variable \(X\) has a uniform distribution, denoted \(U(a,b)\), if its probability density function is:

\(f(x)=\dfrac{1}{b-a}\)

for two constants \(a\) and \(b\), such that \(a<x<b\). A graph of the p.d.f. looks like this:

Uniform PDF

Note that the length of the base of the rectangle is \((b-a)\), while the length of the height of the rectangle is \(\dfrac{1}{b-a}\). Therefore, as should be expected, the area under \(f(x)\) and between the endpoints \(a\) and \(b\) is 1. Additionally, \(f(x)>0\) over the support \(a<x<b\). Therefore, \(f(x)\) is a valid probability density function.

Because there are an infinite number of possible constants \(a\) and \(b\), there are an infinite number of possible uniform distributions. That's why this page is called Uniform Distributions (with an s!) and not Uniform Distribution (with no s!). That said, the continuous uniform distribution most commonly used is the one in which \(a=0\) and \(b=1\).

Cumulative distribution Function of a Uniform Random Variable \(X\)

The cumulative distribution function of a uniform random variable \(X\) is:

\(F(x)=\dfrac{x-a}{b-a}\)

for two constants \(a\) and \(b\) such that \(a<x<b\). A graph of the c.d.f. looks like this:

Uniform CDF

As the picture illustrates, \(F(x)=0\) when \(x\) is less than the lower endpoint of the support (\(a\), in this case) and \(F(x)=1\) when \(x\) is greater than the upper endpoint of the support (\(b\), in this case). The slope of the line between \(a\) and \(b\) is, of course, \(\dfrac{1}{b-a}\).