Midtermfall05 - Midterm 1: Stat 426. Moulinath Banerjee...

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Midterm 1: Stat 426. Moulinath Banerjee University of Michigan October 27, 2005 Announcement: The total number of points is 30 but the maximum you can score is 25. (1) (a) Suppose that the radius of a circle is a random variable having the following probability density function. f ( x ) = 1 8 (3 x + 1) , 0 < x < 2 and 0 otherwise. Determine the probability density function of the area of the circle. (6) (2) Let ( X,Y ) be uniformly distributed inside the ellipse given by x 2 /a 2 + y 2 /b 2 = 1. Thus, the joint density of ( X,Y ) is: f ( x,y ) = 1 π ab 1 x 2 a 2 + y 2 b 2 < 1 ± . where recall that 1 { x 2 /a 2 + y 2 /b 2 < 1 } is the indicator function that is 1 if x 2 /a 2 + y 2 /b 2 < 1 and 0 otherwise. (a) Let ( U,V ) = ( ² 1 X,² 2 Y ) where ² 1 and ² 2 are either 1 or -1.
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