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MIT6_041F10_assn04

# MIT6_041F10_assn04 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 4 Due October 6, 2010 1. Random variables X and Y have the joint PMF c ( x 2 + y 2 ) , if x ∈ { 1 , 2 , 4 } and y ∈ { 1 , 3 } , p X,Y ( x, y ) = 0 , otherwise. (a) What is the value of the constant c ? (b) What is P ( Y < X )? (c) What is P ( Y > X )? (d) What is P ( Y = X )? (e) What is P ( Y = 3)? (f) Find the marginal PMFs p X ( x ) and p Y ( y ). (g) Find the expectations E [ X ], E [ Y ] and E [ XY ]. (h) Find the variances var( X ), var( Y ) and var( X + Y ). (i) Let A denote the event X Y . Find E [ X | A ] and var( X | A ). 2. The newest invention of the 6.041/6.431 staff is a three-sided die with faces numbered 1, 2, and 3. The PMF for the result of any one roll of this die is p X ( x ) = 1 / 2 , if x = 1 , 1 / 4 , if x = 2 , 1 / 4 , if x = 3 , 0 , otherwise. Consider a sequence of six independent rolls of this die, and let X i be the random variable corresponding to the i th roll.

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MIT6_041F10_assn04 - Massachusetts Institute of Technology...

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