Homework 8 - ( y ) . Find the density function of X + Y ....

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550.420 Introduction to Probability Fall 2011 Homework 8 Due Thursday October 27, 2011 1. (2 points) Suppose X and Y have joint density f X,Y ( x, y ) = 1 2 e - y I (0 , ) ( y ) I ( - y,y ) ( x ) Compute P [ X 1 , Y 3]. 2. (2 points) Let X have an exponential distribution with parameter λ and let β be a positive real number. Find the density function of Y = βe X . Remark: The distribution is known as the Pareto distribution with parameters λ and β . 3. (2 points) Suppose X and Y are independent random variables, where X has a Uniform(0,1) distribution and Y has density function f ( y ) = 2 yI (0 , 1)
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Unformatted text preview: ( y ) . Find the density function of X + Y . Hint: Find the distribution function rst. 4. (2 points) Let X and Y be two independent random variables with distribution functions F and G respectively. Find the distribution functions of max { X, Y } and min { X, Y } . 5. (2 points) Let the joint density function of X and Y be given by f ( x, y ) = 8 xyI (0 ,y ) ( xI (0 , 1) ( y ) . Are the random variables independent? Determine if E [ XY ] = E [ X ] E [ Y ]. 1...
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