If w g x and we wish to nd the pdf of w at a

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Unformatted text preview: le problem. displacement of the needle is sin(Θ). Let U denote the vertical distance from the lower endpoint of the needle to the first grid line above that endpoint. It is reasonable to assume that U is uniformly distributed from 0 to 1, and that U is independent of Θ. The needle intersects the grid if and only 1 if U ≤ sin(Θ). The joint density of (Θ, U ), is equal to π over the rectangular region [0, π ] × [0, 1]. Integrating that density over the region {(θ, u) : u ≤ sin(θ)} shown in Figure 4.18 yields u 1 0 " 0 ! Figure 4.18: Region of integration for Buffon needle problem. 142 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES P {needle intersects grid} = P {U ≤ sin(Θ)} π sin θ 1 dudθ π 0 0 π dθ cos(θ) sin(θ) = π π 0 = = π = 0 2 ≈ 0.6366. π The answer can also be written as E [sin(Θ)]. A way to interpret this is that, given Θ, the probability the needle intersects a line is sin(Θ), and averaging over all Θ (in a continuous-type version of the law of total probability) gives the overall answer. Example 4.6.2 Consider the follow...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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