elemprob-fall2010-page36

elemprob-fall2010-page36 - this constant over the set where...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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. An 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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: this constant over the set where X < L cos Θ / 2 and 0 ≤ Θ ≤ π/ 2 and ≤ X ≤ D/ 2. The integral is Z π/ 2 Z L cos θ/ 2 4 πD dxdθ = 2 L πD . If X and Y are independent, then P ( X + Y ≤ a ) = Z Z { x + y ≤ a } f X,Y ( x,y ) dxdy = Z Z { x + y ≤ a } f X ( x ) f Y ( y ) dxdy = Z ∞-∞ Z a-y-∞ f X ( x ) f Y ( y ) dxdy = Z F X ( a-y ) f Y ( y ) dy. Differentiating with respect to a , we have f X + Y ( a ) = Z f X ( a-y ) f Y ( y ) dy. There are a number of cases where this is interesting. (1) If X is a gamma with parameters s and λ and Y is a gamma with parameters t and λ , then straightforward integration shows that X + Y is a 36...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online