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rec12 - X Y can be de-scribed in polar coordinates in terms...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 12 October 19, 2010 1. Show ρ ( aX + b, Y ) = ρ ( X, Y ). 2. Romeo and Juliet have a date at a given time, and each, independently, will be late by amounts of time, X and Y , respectively, that are exponentially distributed with parameter λ . (a) Find the PDF of Z = X - Y by first finding the CDF and then differentiating. (b) Find the PDF of Z by using the total probability theorem. 3. Problem 4.16, page 248 in text. Let X and Y be independent standard normal random variables. The pair (
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Unformatted text preview: X, Y ) can be de-scribed in polar coordinates in terms of random variables R ≥ 0 and Θ ∈ [0 , 2 π ], so that X = R cosΘ , Y = R sinΘ . Show that R and Θ are independent (i.e. show f R, Θ ( r, θ ) = f R ( r ) f Θ ( θ )). (a) Find f R ( r ). (b) Find f Θ ( θ ). (c) Find f R, Θ ( r, θ ). 4. Problem 4.20, page 250 in text. Schwarz inequality . Show that for any random variables X and Y , we have ( E [ XY ]) 2 ≤ E [ X 2 ] E [ Y 2 ] . Page 1 of 1...
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