14.7 - Uniform Properties
14.7 - Uniform PropertiesHere, we present and prove three key properties of a uniform random variable.
Theorem
The mean of a continuous uniform random variable defined over the support \(a<x<b\) is:
\(\mu=E(X)=\dfrac{a+b}{2}\)
Proof
Theorem
The variance of a continuous uniform random variable defined over the support \(a<x<b\) is:
\(\sigma^2=Var(X)=\dfrac{(b-a)^2}{12}\)
Proof
Because we just found the mean \(\mu=E(X)\) of a continuous random variable, it will probably be easiest to use the shortcut formula:
\(\sigma^2=E(X^2)-\mu^2\)
to find the variance. Let's start by finding \(E(X^2)\):
Now, using the shortcut formula and what we now know about \(E(X^2)\) and \(E(X)\), we have:
\(\sigma^2=E(X^2)-\mu^2=\dfrac{b^2+ab+a^2}{3}-\left(\dfrac{b+a}{2}\right)^2\)
Simplifying a bit:
\(\sigma^2=\dfrac{b^2+ab+a^2}{3}-\dfrac{b^2+2ab+a^2}{4}\)
and getting a common denominator:
\(\sigma^2=\dfrac{4b^2+4ab+4a^2-3b^2-6ab-3a^2}{12}\)
Simplifying a bit more:
\(\sigma^2=\dfrac{b^2-2ab+a^2}{12}\)
and, finally, we have:
\(\sigma^2=\dfrac{(b-a)^2}{12}\)
as was to be proved.
Theorem
The moment generating function of a continuous uniform random variable defined over the support \(a < x < b\) is:
\(M(t)=\dfrac{e^{tb}-e^{ta}}{t(b-a)}\)