Raising the bar

A little creativity to explain the uniform distribution

They say: (Agresti and Franklin, Exercise 6.12) Uniform distribution A random number generator is used to generate a real number between 0 and 1, equally likely to fall anywhere in this interval of values. (For instance, 0.3794259832... is a possible outcome.)
a. Sketch a curve of the probability distribution of this random variable, which is the continuous version of the uniform distribution (see Exercise 6.1).
b. What is the mean of this probability distribution?
c. Find the probability that this random variable falls between 0.25 and 0.75.

I say: Using a random number generator to introduce a continuous uniform distribution is a wild idea, given the target audience of this book. I do the following. I hold my computer cable and say: Let's consider a piece of this cable. Do you think the probability of this cable breaking here (indicating a point on the piece) is higher than there (indicating another point)? Everybody says: No. Then I propose the students to sketch the density. Let's denote A the left point of this stretch and B the right point. The students tell me that between A and B the density is constant. And to the left of A? Zero. How about to the right of B? It is also zero. Then we establish the value of the constant using the property that the total area under the density should be one. This involves calculation of the area of a rectangle - anybody can do this.

Prompt your students to use densities to compare different situations. Here is a good exercise from my book:

Take two related continuous random variables and draw on the same graph what you think would be their densities. For example, you can take:
(i) distributions of income in a wealthy neighborhood and in a poor neighborhood,
(ii) distributions of temperature in winter and summer in a given geographic location;
(iii) distributions of electricity consumption in two different locations at the same time of the year.
Usually when doing this exercise, the students forget that the total density is 1.