Chapter 2 Continuous Probability Densities Simulation of Continuous Probabilities In this section we shall show how we can use computer simulations for experiments that have a whole continuum of possible outcomes. Example We begin by constructing a spinner, which consists of a circle of unit circumference | Chapter 2 Continuous Probability Densities Simulation of Continuous Probabilities In this section we shall show how we can use computer simulations for experiments that have a whole continuum of possible outcomes. Probabilities Example We begin by constructing a spinner which consists of a circle of unit circumference and a pointer as shown in Figure . We pick a point on the circle and label it 0 and then label every other point on the circle with the distance say x from 0 to that point measured counterclockwise. The experiment consists of spinning the pointer and recording the label of the point at the tip of the pointer. We let the random variable X denote the value of this outcome. The sample space is clearly the interval 0 1 . We would like to construct a probability model in which each outcome is equally likely to occur. If we proceed as we did in Chapter 1 for experiments with a finite number of possible outcomes then we must assign the probability 0 to each outcome since otherwise the sum of the probabilities over all of the possible outcomes would not equal 1. In fact summing an uncountable number of real numbers is a tricky business in particular in order for such a sum to have any meaning at most countably many of the summands can be different than 0. However if all of the assigned probabilities are 0 then the sum is 0 not 1 as it should be. In the next section we will show how to construct a probability model in this situation. At present we will assume that such a model can be constructed. We will also assume that in this model if E is an arc of the circle and E is of length p then the model will assign the probability p to E. This means that if the pointer is spun the probability that it ends up pointing to a point in E equals p which is certainly a reasonable thing to expect. 41 42 CHAPTER 2. CONTINUOUS PROBABILITY DENSITIES Figure A spinner. To simulate this experiment on a computer is an easy matter. Many computer software packages .
