# The determinant of the jacobian is detj det 2u 0 1v

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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.

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