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Unformatted text preview: ndependent of its position
relative to the vertical grid. Let U be as in the solution to Buﬀon’s needle problem, and let V
similarly denote the distance from the leftmost endpoint of the needle to the ﬁrst vertical grid
line to the right of that point, as shown in Figure 4.19. Then U and V are independent, and the sin(! ) U !
V
cos(!) Figure 4.19: Variation of the Buﬀon needle problem, with horizontal and vertical grids.
needle misses both grids if and only if U ≥ sin() and V ≥  cos(Θ). Therefore, P (Mh Mv Θ =
θ) = P {U ≥ sin(θ), V ≥  cos(θ)} = P {U ≥ sin(θ)}P {V ≥  cos(θ)} = (1 − sin(θ))(1 −  cos(θ)). 4.6. ADDITIONAL EXAMPLES USING JOINT DISTRIBUTIONS 143 Averaging over Θ using its pdf yields (using the trigometric identity 2 sin(θ) cos(θ) = sin(2θ))
π 1
π P ( Mh Mh ) = (1 − sin(θ))(1 −  cos(θ))dθ
0
π /2 2
π = (1 − sin(θ))(1 − cos(θ))dθ
0
π /2 2
π = 1 − sin(θ) − cos(θ) − sin(θ) cos(θ)dθ
0
π /2 2
π
2
π =
= 1 − sin(θ) − cos(θ) −
0 π
1
−1−1+
2
2 =1− sin(2θ)
dθ
2 3
≈ 0.045.
π The true probabilit...
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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 Institute of Technology.
 Spring '08
 Zahrn
 The Land

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