# homework5 - IE 300 GE 331 Spring 2009 Homework#5 Due April...

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IE 300 / GE 331 Spring 2009 Homework #5 Due: April 9th Problem 1. Random variables X and Y have the joint PDF shown below: (a) Prepare neat, fully labeled sketches of f X | Y ( u | v ). (b) Find E [ X | Y = v ] and var[ X | Y = v ]. (c) Find E [ X ]. Problem 2. X and Y denote independent standard Gaussian random variables. (a) What is the joint pdf f X , Y ( u,v ) of X and Y ? 1

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IE 300 / GE 331: Homework #5 2 (b) Sketch the u - v plane and indicate on it the region over which you need to integrate the joint pdf in order to ﬁnd P {X 2 + Y 2 > 2 α 2 } . Compute P {X 2 + Y 2 > 2 α 2 } . (c) Let Z = X 2 + Y 2 . What is the pdf of Z ? (d) Express P {|X| > α } in terms of the complementary unit Gaussian CDF func- tion Q ( x ), and use this to write P {|X| > α, |Y| > α } in terms of Q ( x ). (Re- member commas mean intersections). (e) On your sketch of part (b), show the region over which you must integrate the joint pdf to ﬁnd P {|X| > α, |Y| > α, } . Use your sketch to prove the following
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homework5 - IE 300 GE 331 Spring 2009 Homework#5 Due April...

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