homework5

homework5 - c. {X 2 < Y}. 7. Let X and Y be...

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C OMER ECE 600 Homework 5 1. Let the joint cumulative distribution function (cdf) of random variables X and Y be F XY ( x,y ). Show that ) , ( F ) , ( F ) , ( F ) , ( F }) Y { } X P({ 1 1 XY 1 2 XY 2 1 XY 2 2 XY 2 1 2 1 y x y x y x y x y y x x + - - = < < 2. Determine the constant b such that each of the following is a valid probability density function (pdf). a. ± ² = elsewhere 0 0 ; 1 0 3 ) , ( f XY b y x xy y x b. ± ² - = elsewhere 0 1 0 ; 5 . 0 0 ) 1 ( ) , ( f XY y x y bx y x c. ± ² < < + = elsewhere 0 2 0 ; 1 0 ) 4 ( ) , ( f 2 2 XY y x y x b y x 3. If f XY ( x,y ) = g( x )h( y ), find the marginal pdfs f X ( x ) and f Y ( y ). 4. For two independent continuous random variables X and Y, show that dx x x dx x x ) ( f ) ( F 1 ) ( f ) ( F X}) P({Y Y X X Y ³ ³ - - - = = 5. Suppose f XY ( x , y ) is uniform with region of support shown in the figure below. Find f X ( x ) and f Y ( y ). Are X and Y independent?
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6. The random vector (X,Y) has a joint pdf ± ² > > = - - otherwise 0 0 , 0 e e 2 ) , ( f 2 XY y x y x y x Find the probability of the following events: a. {X + Y 8}. b. {X < Y}.
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Unformatted text preview: c. {X 2 &lt; Y}. 7. Let X and Y be independent random variables with Y uniformly distributed from 0 to 1. Let X have cdf and pdf F X ( x ) and f X ( x ), respectively. Show that the pdf of the random variable Z = X + Y is given by ) 1 ( F ) ( F ) ( f X X Z--= z z z 8. The joint density function of X and Y is given by &amp; &gt; &gt; = +-otherwise , ) , ( f ) ( y x e y x y x XY Find the density function of the random variable X/Y 9. Two fair dice are rolled. Find the joint probability mass function of X and Y when a. X is the larger value rolled and Y is the sum of the two values. b. X is the smaller and Y is the larger value rolled....
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homework5 - c. {X 2 &amp;amp;lt; Y}. 7. Let X and Y be...

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