# 14.7 - Uniform Properties

14.7 - Uniform Properties

Here, 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:

$$\mu=E(X)=\dfrac{a+b}{2}$$

## 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:

$$M(t)=\dfrac{e^{tb}-e^{ta}}{t(b-a)}$$

### Proof

 [1] Link ↥ Has Tooltip/Popover Toggleable Visibility