11.2: The Uniform Continuous Distribution
As mentioned in Week 9, there are certain processes which occurs more frequently than others and so we will attach "brand" names to the random variables which models these processes. In terms of discrete random variables, we studied the (discrete) Uniform Random Variable, The Bernoulli and Binomial Random Variables, The Geometric and Negative Binomial Random Variables, and the Poisson Random Variable.
For the class of continuous random variables, we will study the (continuous) Uniform Random Variable, the Exponential Random Variable, and the Normal Random Variable. We first begin with the (continuous) Uniform Random Variable.
Definition: Let \( a < b \) be real numbers. Suppose \( X \) is a random variable who has the following density:
\[
f(x) =
\begin{cases}
\frac{1}{b-a}, & \text{if} ~ a \leq x \leq b \ \\
0 , & \text{otherwise} \\
\end{cases}
\]
Then we say \( X \) has the uniform distribution on the interval \( [a,b] \) and we write \( X \sim \mathcal{U} [ a , b ] \).
Notice that the (continuous) Uniform Random Variable is something which is not totally new for us. Observe that we actually studied the \(\mathcal{U} [ 4, 200 ] \) random variable in the last section and so many of the proofs are left to the reader. Additionally, recall that the point of a random variable is to model some sort of quantity under certain assumptions and so we can ask, "what does the (continuous) Uniform random variable model?"
The (continuous) Uniform Random Variable models the outcome of an experiment where the experiment is choosing a number at random from the interval \( [a,b] \).
We end this section with a theorem concerning some of the properties of a discrete Uniform Random Variable. Their proofs are left as homework exercises.
Theorem: If \( X \sim \mathcal{U} [ a , b ] \) then:
\begin{align*} \mathbb{E}[X] &= \frac{a+b}{2} \\ \mathbb{V}ar[X] &= \frac{(b-a)^2}{12} \\ F(x) &= \begin{cases}
0, & \mbox{if } x < a \\ \frac{x-a}{b-a}, & \mbox{if } a \leq x < b \\ 1, & \mbox{if } x \geq b
\end{cases} \end{align*}