9.2: The Uniform Random Variable
For the past few weeks, we have been discussing random variables. In this section, we will begin to discuss important random variables and their distributions. Before we do so, allow us to ask a question: what is the point of a random variable? Or rather, why should we study random variables?
The answer is the point of a random variable is to model some sort of real life process or phenomenon and this act of modeling a quantity is often the first step towards predicting or forecasting the quantity of interest.
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. Specifically, we will study the Uniform Random Variable, The Bernoulli and Binomial Random Variable, The Geometric and Negative Binomial Random Variable, and the Poisson Random Variable.
We should note that there are infinitely many random variables out there in the universe and so it would be impossible to discuss every such random variable. For that reason, it is nice to have a framework to follow whenever you are introduced to a random variable in your future courses. Here is that framework which we will apply to he Uniform Random Variable, The Bernoulli and Binomial Random Variable, The Geometric and Negative Binomial Random Variable, and the Poisson Random Variable.
1) We should first ask how is the probability mass function or cumulative distribution function defined.
2) We should then realize that the point of a random variable is to model some sort of quantity. Hence, we then ask ourselves what does this random variable models and under what assumptions are required? Essentially, we should try to interpret the random variable.
3) When feasible, we should show that the assumptions in (2) yield the probability mass function or cumulative distribution function outlined in (1).
With that said, allow us to apply the framework to our first brand name random variable, The Uniform Random Variable. We first seek to understand how The Uniform Random Variable is defined.
Definition: Let \( a \leq b \) be integers. A random variable \(X\) whose probability mass function is given by \[ f(x) = P(X = x) =
\begin{cases}
\frac{1}{b-a+1}, & \mbox{if } x = a, a+1, a+2, \ldots, b \\
0, & \mbox{otherwise}
\end{cases}\]
is said to have the Uniform Distribution on the integers \(a, a+1, a+2, \ldots, b\) and we write \(X \sim \mathcal{U} \{a,b\} \) which is read as " \(X\) is distributed as a Uniform Random Variable on the integers \( a, a+1, a+2, \ldots, b\)".
Again, the point of a random variable is to model some sort of quantity under certain assumptions and so we can ask, "what does the above random variable model and why?" We first answer what does the discrete Uniform Random Variable models.
The (discrete) Uniform Random Variable models the outcome of an experiment where the experiment is choosing an integer from \(a, \ldots, b\) at random.
We now show why the experiment outlined above yields the discrete Uniform Random Variable.
Theorem: If \(X\) models the outcome of an experiment where the experiment is choosing an integer from \(a, \ldots, b\) at random, then \(X \sim \mathcal{U} \{a,b\} \).
- Proof:
-
To prove this, we simply derive the probability mass function as discussed in Section 8.1 . We first ask ourselves what are the possible values of \(X\)? We see that the integer we can select is either \(a, a+1, a+2, \ldots\) or \( b\) and so \( X: a, a+1, a+2, \ldots, b\). We now must find the probability that \(X\) takes each value.
\begin{align*} P(X=a) &= P(\{ \text{we selected the number} ~ a \}) \underbrace{=}_{SSS} \frac{| \{ \text{we selected} ~ a \} | }{|S|} = \frac{1}{b-a+1} \\ P(X= a+1) &= P( \{ \text{we selected the number} ~ a+1 \} ) \underbrace{=}_{SSS} \frac{ | \{ \text{we selected} ~ a+1 \} |}{|S|} = \frac{1}{b-a+1} \\ \ldots \\ \ldots \\ \ldots \\ P(X=b) &= P( \{ \text{we selected the number} ~ b \} ) \underbrace{=}_{SSS} \frac{| \{ \text{we selected} ~ b \} | }{|S|} = \frac{1}{b-a+1} \end{align*}
Hence, \( f(x) = \begin{cases}
\frac{1}{b-a+1}, & \mbox{if } x = a, a+1, a+2, \ldots, b \\
0, & \mbox{otherwise}
\end{cases}\ \nonumber\ \) so \(X \sim \mathcal{U} \{a,b\} \).
We end this section with a theorem concerning some of the properties of a discrete Uniform Random Variable.
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+1)^2 - 1}{12} \\ F(x) &= \begin{cases}
0, & \mbox{if } x < a \\ \frac{ \lfloor x \rfloor -a+1}{b-a+1}, & \mbox{if } a \leq x < b \\ 1, & \mbox{if } x \geq b
\end{cases} \end{align*}