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Probability: An Introduction with Statistical Applications, 2nd Edition by John J. Kinney

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Chapter 3Continuous Random Variables and Probability Distributions

3.1 Introduction

Discrete random variables were discussed in Chapter 2. However, it is not always possible to describe all the possible outcomes of an experiment with a finite, or countably infinite, sample space. As an example, consider the wheel shown in Figure 3.1 where the numbers from 0 to 1 have been marked on the outside edge.

c03f001

Figure 3.1 The spinner.

The experiment consists of spinning the spinner and recording where the arrow stops. It would be natural here to consider the sample space, c03-math-0001, to be

equation

c03-math-0003 is infinite, but not countably infinite.

Now the question arises, “What probability should be put on each of the points in c03-math-0004?” Surely, if the wheel is fair, each point should receive the same probability and the total probability should be 1. What value should that probability be?

Suppose, for the sake of argument, that a probability of 0.0000000000000000000001 = 10−22 is put on each point. It is easy to show that ...

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