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Unformatted text preview: SYSC 4005/5001 Assignment # 1 Problem 1 (Buffon’s Needle): This problem presents a static simulation for the computation of number π . The method was first suggested by C. de Buffon in 1777. Imagine an infinite plane covered with a grid of parallel lines at a distance 2a from each other. A line segment (i.e. a needle) of length 2b (b ≤ a) is dropped at random on the plane. The position of the needle on the plane is determined by the following two random variables: A) Distance Y between the centre of the needle and the nearest line. Obviously Y is uniformly distributed in (0, a). B) The angle Φ between the needle and the line. Again we assume that the angle is uniformly distributed in (0, π /2). The random variables Y, Φ are assumed to be independent. The needle crosses a line when Y ≤ b sin Φ . Q1: Show that the needle crosses a line with probability p where π a b b Y P p 2 ) sin ( = Φ ≤ = According to the law of large numbers if the segment is dropped N times and on n of those occasions it crosses a line, then p N n N = ∞ → lim with probability one. Therefore an approximation of π can be evaluated numerically through the following simulation: 1) Select values for a and b....
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- Winter '10
- Statistics, Probability theory