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elemprob-page21 - One can conclude from this that fX,Y(x y...

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One can conclude from this that f X,Y ( x, y ) = f X ( x ) f Y ( y ) , or again the joint density factors. Going the other way, one can also see that if the joint density factors, then one has independence. Example. Suppose one has a floor made out of wood planks and one drops a needle onto it. What is the probability the needle crosses one of the cracks? Suppose the needle is of length L and the wood planks are D across. Answer. Let X be the distance from the midpoint of the needle to the nearest crack and let Θ be the angle the needle makes with the vertical. Then X and Θ will be independent. X is uniform on [0 , D/ 2] and Θ is uniform on [0 , π/ 2]. A little geometry shows that the needle will cross a crack if L/ 2 > X/ cos Θ. We have f X, Θ = 4 πD and so we have to integrate this constant over the set where X < L cos Θ / 2 and 0 Θ π/ 2 and 0 X D/ 2. The integral is π/ 2 0 L cos θ/ 2 0 4 πD dx dθ = 2 L πD . If X and Y are independent, then P ( X + Y a ) = { x + y a } f X,Y ( x, y ) dx dy = { x + y a } f X ( x ) f Y ( y ) dx dy = -∞ a -
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