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set8 - ECSE-305 Winter 2009 Probability and Random Signals...

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ECSE-305, Winter 2009 Probability and Random Signals I Assignment #8 Posted: Thursday, March 19, 2009. Due: Thursday, March 26, 2009, 2h30pm. Important notes: Assignments without this cover page will be discarded. Student #1: Name: ID: Student #2: Name: ID: Question Marks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Total
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1. The joint PDF of random variables X and Y is given by f ( x, y ) = braceleftBigg cx k y l , 0 x 1 , 0 y 1 0 , otherwise where k and l are positive integers. (a) Find an expression for the constant c in terms of k and l (b) Find a necessary condition on the values of k and l such that P ( Y > X ) > 1 / 2. (c) Find the marginal PDFs of X and Y . 2. Using an appropriate software package (e.g. Matlab), sketch the 3D graph of the joint normal PDF N ( μ X , μ Y , ρ, σ 2 X , σ 2 Y ) for the following parameter values (note: hand-drawing will not be accepted): (a) μ X = μ Y = 0, σ X = σ Y = 1 and ρ = 0,; (b) μ X = 1 , μ Y = 1 / 2, σ X = σ Y = 1 and ρ = 0,; (c) μ X = μ Y = 0, σ X = 1 , σ Y = 2 and ρ = 0,; (d) μ X = μ Y = 0, σ X = σ Y = 1 and ρ = 0 . 5.
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