The Uniform Distribution

A graphic of a Uniform distribution. Note that the distribution is symmetric. It does not have a mode, as all outcomes are equally likely.

In the previous chapter, we covered several discrete distribuions. The most important of these is the Binomial distribution; it help in modeling discrete random variables. In this chapter, we cover several continuous distributions. The first and the simplest is the Uniform distribution. It is easy to work with, because calculating probabilities (area) reduces to just calculating areas of rectangles.

The Uniform Distribution

Section 6.2 starts off with the continuous Uniform distribution. The book uses this distribution to help introduce the probability density function (pdf). When you have a pdf, the probabilities are just areas under the pdf curve. Notice that Example 6.1 uses the shape of the Uniform’s pdf to calculate areas (probabilities).

For those who are interested, here is the functional form of the pdf for the Uniform distribution. Note it is very simple.

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

Here, $a$ is the minimum value and $b$ is the maximum value. In Example 6.1 in the book (Elevator Waiting Times), $a = 0$ and $b=4$ minutes. Since the pdf does not depend on $x$, its height is constant. That means we can calculate areas (probabilities) simply, because the area of a rectangle is just height × width.

Examples

To help you better understand the Uniform distribtion, here are a few examples. They all rely on the Elevator Waiting Times example from the textbook, Page 292.

A. What is the probability of having to wait no more than 2 minutes?
The first step is to translate the words above into probability terms. Here, we are given X ~ UNIF(a=0; b=4). We are asked to calculate P[X ≤ 2].
Ans. =(2-0)/(4-0) = 0.500

B. What is the probability of having to wait no more than 1 minute?
Again, the first step is to translate the words above into probability terms. Here, we are given X ~ UNIF(a=0; b=4). We are asked to calculate P[X ≤ 1].
Ans. =(1-0)/(4-0) = 0.250

C. What is the probability of having to wait more than 1 minute?
Again, we are given X ~ UNIF(a=0; b=4). We are asked to calculate P[X > 1]. This is equal to 1 − P[X ≤ 1]
Ans. =1-(1-0)/(4-0) = 0.750

Videos

These videos cover the theory behind the Uniform distribution using Excel.

In addition to these videos, there are other videos on YouTube for probabilities of a Uniform distribtion in Excel. The following search link will take you to YouTube and provide you with a non-exhaustive list: Uniform probabilities in Excel.

That is it. In this mini-lecture, we looked at the Uniform distribution and what it can be used for. We saw how to perform probability calculations using Excel. Without question, the Uniform distribution is the easiest continuous distribution we have. The next distribution covered is the Normal distribution. While calculations are more difficult, it is much more useful in modeling real life.