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📚 Statistics with Technology 2e
Chapters ▾
1 Statistical Basics2 Graphical Descriptions of Data3 Examining the Evidence Using Graphs and Statistics4 Probability5 Discrete Probability Distributions6 Continuous Probability Distributions7 One-Sample Inference8 Estimation9 Two-Sample Inference10 Regression and Correlation11 Chi-Square and ANOVA Tests12 Appendix: Critical Value Tables
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6.1 Uniform Distribution

If you have a situation where the probability is always the same, then this is known as a uniform distribution. An example would be waiting for a commuter train. The commuter trains on the Blue and Green Lines for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during peak hours of ten minutes ("2012 annual report," 2012). If you are waiting for a train, you have anywhere from zero minutes to ten minutes to wait. Your probability of having to wait any number of minutes in that interval is the same. This is a uniform distribution. The graph of this distribution is in Figure 1.

A bar chart showing a uniform probability distribution of 0.1 for train waiting times from 0 through 10 minutes.
Figure

Suppose you want to know the probability that you will have to wait between five and ten minutes for the next train. You can look at the probability graphically such as in Figure 2.

A bar chart of a uniform waiting-time distribution, shading probability density 0.1 for waiting times from 5 to 10 minutes.
Figure 2: Uniform Distribution with P (5<x<10)

How would you find this probability? Calculus says that the probability is the area under the curve. Notice that the shape of the shaded area is a rectangle, and the area of a rectangle is length times width. The length is 10−5=5 and the width is 0.1. The probability is P(5<x<10)=0.1*5=0.5, where and x is the waiting time during peak hours.

Notice that in Example 1d, the probability is equal to one. This is because the probability that was computed is the area under the entire curve. Just like in discrete probability distributions, where the total probability was one, the probability of the entire curve is one. This is the reason that the height of the curve is 0.1. In general, the height of a uniform distribution that ranges between a and b, is 1b−a.

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