# therefore proposition 474 yields that for r in the

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Unformatted text preview: y of missing both grids is almost three times smaller than what it would be if Mh and Mv were independent. Intuitively, the events are negatively correlated. That is because if Mh is true, it is likely that the position of the needle is more horizontal than vertical, and that makes it less likely that Mv is true. A way to express the negative correlation is to note that (h 0.045 P (Mv |Mh ) = P PMMM)h ) ≈ 0.363 ≈ 0.124, which is nearly three times smaller than the unconditional (h probability, P (Mv ) ≈ 0.363. √ Example 4.6.3 Suppose X and Y are jointly continuous-type random variables. Let R = X 2 + Y 2 , so R is the distance of the random point (X, Y ) from the origin. Express fR in terms of fX,Y . Solution: Clearly R is a nonnegative random variable. We proceed to ﬁnd its CDF. Let c > 0, and let D(c) denote the disk of radius c centered at the origin. Using polar coordinates, FR (c) = P {R ≤ c} = P {(X, Y ) ∈ D(c)} = fX,Y (u, v )dudv D(c) 2π c = fX,Y (r cos(θ), r sin(θ))rd...
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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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