Chapter 5: Continuous Random Variables
5.2 The Uniform Distribution
Learning Objectives
At the end of this section, students will be able to:
- describe uniform probability distribution and apply it appropriately.
The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.
Example
The data in the table below are 55 smiling times, in seconds, of an eight-week-old baby.
| 10.4 | 19.6 | 18.8 | 13.9 | 17.8 | 16.8 | 21.6 | 17.9 | 12.5 | 11.1 | 4.9 |
| 12.8 | 14.8 | 22.8 | 20.0 | 15.9 | 16.3 | 13.4 | 17.1 | 14.5 | 19.0 | 22.8 |
| 1.3 | 0.7 | 8.9 | 11.9 | 10.9 | 7.3 | 5.9 | 3.7 | 17.9 | 19.2 | 9.8 |
| 5.8 | 6.9 | 2.6 | 5.8 | 21.7 | 11.8 | 3.4 | 2.1 | 4.5 | 6.3 | 10.7 |
| 8.9 | 9.4 | 9.4 | 7.6 | 10.0 | 3.3 | 6.7 | 7.8 | 11.6 | 13.8 | 18.6 |
The sample mean = 11.49 and the sample standard deviation = 6.23.
We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. This means that any smiling time from zero to and including 23 seconds is equally likely. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.
Let X = length, in seconds, of an eight-week-old baby’s smile.
The notation for the uniform distribution is
X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.
The probability density function is
f(x) = [latex]\frac{1}{b-a}[/latex] for a ≤ x ≤ b.
For this example, X ~ U(0, 23) and f(x) = [latex]\frac{1}{23-0}[/latex] for 0 ≤ X ≤ 23.
Formulas for the theoretical mean and standard deviation are
[latex]\mu =\frac{a+b}{2}[/latex] and [latex]\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}[/latex]
For this problem, the theoretical mean and standard deviation are
μ = [latex]\frac{0\text{ }+\text{ }23}{2}[/latex]
= 11.50 seconds and σ = [latex]\sqrt{\frac{{\left(23\text{ }-\text{ }0\right)}^{2}}{12}}[/latex]
= 6.64 seconds.
Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example.
Your Turn!
The data that follow are the number of passengers on 35 different charter fishing boats. The sample mean = 7.9 and the sample standard deviation = 4.33. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. State the values of a and b. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation.
| 1 | 12 | 4 | 10 | 4 | 14 | 11 |
| 7 | 11 | 4 | 13 | 2 | 4 | 6 |
| 3 | 10 | 0 | 12 | 6 | 9 | 10 |
| 5 | 13 | 4 | 10 | 14 | 12 | 11 |
| 6 | 10 | 11 | 0 | 11 | 13 | 2 |
Solution
a is zero; b is 14; X ~ U (0, 14); μ = 7 passengers; σ = 4.04 passengers
Example
a. Refer to the data from the first example (copied below). What is the probability that a randomly chosen eig