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Unformatted text preview: 2.1. SIMULATION OF CONTINUOUS PROBABILITIES 43 1 x 1 y y = x 2 E Figure 2.2: Area under y = x 2 . for this simple region we can find the exact area by calculus. In fact, Area of E = 1 x 2 dx = 1 3 . We have remarked in Chapter 1 that, when we simulate an experiment of this type n times to estimate a probability, we can expect the answer to be in error by at most 1 / √ n at least 95 percent of the time. For 10,000 experiments we can expect an accuracy of 0.01, and our simulation did achieve this accuracy. This same argument works for any region E of the unit square. For example, suppose E is the circle with center (1 / 2 , 1 / 2) and radius 1/2. Then the probability that our random point ( x, y ) lies inside the circle is equal to the area of the circle, that is, P ( E ) = π 1 2 2 = π 4 . If we did not know the value of π , we could estimate the value by performing this experiment a large number of times!...
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- Spring '09