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Quiz 2 - Solutions

# Quiz 2 - Solutions - No Since f y | x 6 = f y or f x,y 6 =...

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STATISTICS 321 – Spring 2009 Name: Quiz 2 (20 points) – This quiz has 2 sides. 1. Let X and Y be two continuous random variables with joint probability density function given by f ( x, y ) = ( x + y, 0 x 1 , 0 y 1 0 , otherwise (a) Find the marginal probability density function of X , that is, f ( x ). f ( x ) = Z 1 0 f ( x, y ) dy = " xy + y 2 2 # y =1 y =0 = ( x + 1 2 , 0 x 1 0 , otherwise (b) Find the marginal probability density function of Y , that is, f ( y ). f ( y ) = Z 1 0 f ( x, y ) dx = " x 2 2 + xy # x =1 x =0 = ( y + 1 2 , 0 y 1 0 , otherwise (c) Find the conditional probability density function of Y given X = 0 . 25, that is, f ( y | x = 0 . 25). f ( y | x = 0 . 25) = f ( x = 0 . 25 , y ) f ( x = 0 . 25) = 0 . 25 + y 0 . 25 + 0 . 5 = ( 4 y +1 3 , 0 y 1 0 , otherwise (d) Are X and Y independent random variables? Why or why not?
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Unformatted text preview: No. Since f ( y | x ) 6 = f ( y ) or f ( x,y ) 6 = f ( x ) .f ( y ) for all pairs of x and y . (e) TRUE / FALSE : Covariance( X,Y ) = 0 2. For two random variables W and Z , which of the following is/are true? (a) V ar ( W-Z ) = V ar ( W + Z ) always (b) V ar ( W-Z ) 6 = V ar ( W + Z ) always (c) V ar ( W-Z ) = V ar ( W + Z ) if W and Z are independent of each other (d) V ar ( W-Z ) = V ar ( W + Z ) ± constant...
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