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Unformatted text preview: 2.1. SIMULATION OF CONTINUOUS PROBABILITIES 45 d 1/2 θ Figure 2.4: Buffon’s experiment. θ 1/2 d π /2 E Figure 2.5: Set E of pairs ( θ, d ) with d < 1 2 sin θ . Now the area of the rectangle is π/ 4, while the area of E is Area = π/ 2 1 2 sin θ dθ = 1 2 . Hence, we get P ( E ) = 1 / 2 π/ 4 = 2 π . The program BuffonsNeedle simulates this experiment. In Figure 2.6, we show the position of every 100th needle in a run of the program in which 10,000 needles were “dropped.” Our final estimate for π is 3.139. While this was within 0.003 of the true value for π we had no right to expect such accuracy. The reason for this is that our simulation estimates P ( E ). While we can expect this estimate to be in error by at most 0.001, a small error in P ( E ) gets magnified when we use this to compute π = 2 /P ( E ). Perlman and Wichura, in their article “Sharpening Buffon’s 46 CHAPTER 2. CONTINUOUS PROBABILITY DENSITIES 0.00 5.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.005....
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This note was uploaded on 09/15/2009 for the course SCF scf taught by Professor Scf during the Spring '09 term at Indian Institute Of Management, Ahmedabad.
- Spring '09