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 ay∞ f X ( x ) f Y ( y ) dxdy = Z F X ( ay ) f Y ( y ) dy. Diﬀerentiating with respect to a , we have f X + Y ( a ) = Z f X ( ay ) 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...
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 Spring '10
 ansan
 Factors, dx dy, joint density factors

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