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Unformatted text preview: Stat 230 - Assignment 3 Due in class on Friday, December 3, 2010 1. A continuous random variable X is said to have the Gamma distribution with parameters > 0 and > 0 if f ( x ) = ( x - 1 ( ) e- x x otherwise . Here, ( ) = R x - 1 e- x d x is the Gamma function introduced in class. (a) Show that f ( . ) is a legitimate Probability Density Function. (b) Use the properties of the Gamma function to obtain E( X ) and Var( X ). (c) Verify that setting = 1 results in the exponential distribution with parameter = 1 . 2. A random variable X is said to have a Cauchy distribution with parameter > 0 if f X ( x ) = ( 2 + x 2 ) , x R . (a) Show that E( X ) and Var( X ) both do not exist. (b) Let Y = 1 X . Show that Y has a Cauchy distribution with parameter 1 . (c) Find the c.d.f. F ( . ) and the inverse c.d.f. F- 1 ( . ) for the random variable X ....
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- Fall '10