tailieunhanh - Introduction to Probability phần 2

Chúng tôi đã nhận xét trong chương 1, khi chúng tôi mô phỏng một thí nghiệm của loại hình này n lần để ước tính xác suất, chúng ta có thể mong đợi câu trả lời cho được do lỗi của tại √ nhất 1 / n ít nhất 95% thời gian. Cho thí nghiệm, chúng tôi có thể tính chính xác của 0,01, và mô phỏng của chúng tôi đã đạt được độ chính xác này. | . SIMULATION OF CONTINUOUS PROBABILITIES 43 y Figure Area under y x2. for this simple region we can find the exact area by calculus. In fact Area of E I x2 dx - . Jo 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 ựn at least 95 percent of the time. For 10 000 experiments we can expect an accuracy of 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 n 2 2 4. If we did not know the value of n we could estimate the value by performing this experiment a large number of times The above example is not the only way of estimating the value of n by a chance experiment. Here is another way discovered by 1G. L. Buffon in Essai d Arithmetique Morale Oeuvres Completes de Buffon avec Supple- ments tome iv ed. Dumenil Paris 1836 . 44 CHAPTER 2. CONTINUOUS PROBABILITY DENSITIES Figure Computing the area by simulation. Buffon s Needle Example Suppose that we take a card table and draw across the top surface a set of parallel lines a unit distance apart. We then drop a common needle of unit length at random on this surface and observe whether or not the needle lies across one of the lines. We can describe the possible outcomes of this experiment by coordinates as follows Let d be the distance from the center of the needle to the nearest line. Next let L be the line determined by the needle and define 6 as the acute angle that the line L makes with the set of parallel lines. The reader should certainly be wary of this description of the sample space. We are attempting to coordinatize a set of line segments. To see why one must be careful in the choice of coordinates see Example . Using this