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Chapter2

# Chapter2 - Chapter 2 Continuous Probability Densities 2.1...

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Chapter 2 Continuous Probability Densities 2.1 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 2.1 We begin by constructing a spinner, which consists of a circle of unit circumference and a pointer as shown in Figure 2.1. 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

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42 CHAPTER 2. CONTINUOUS PROBABILITY DENSITIES 0 x Figure 2.1: A spinner. To simulate this experiment on a computer is an easy matter. Many computer software packages have a function which returns a random real number in the in- terval [0 , 1]. Actually, the returned value is always a rational number, and the values are determined by an algorithm, so a sequence of such values is not truly random. Nevertheless, the sequences produced by such algorithms behave much like theoretically random sequences, so we can use such sequences in the simulation of experiments. On occasion, we will need to refer to such a function. We will call this function rnd . 2 Monte Carlo Procedure and Areas It is sometimes desirable to estimate quantities whose exact values are difficult or impossible to calculate exactly. In some of these cases, a procedure involving chance, called a Monte Carlo procedure , can be used to provide such an estimate.
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