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A3 - Stat 230 Assignment 3 Due in class on Friday December...

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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 α > 0 and β > 0 if f ( x ) = ( x α - 1 Γ( α ) β α e - x β x 0 0 otherwise . Here, Γ( α ) = R 0 x α - 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 α = 1 results in the exponential distribution with parameter λ = 1 β . 2. A random variable X is said to have a Cauchy distribution with parameter α > 0 if f X ( x ) = α π ( α 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 α . (c) Find the c.d.f. F ( . ) and the inverse c.d.f. F - 1 ( . ) for the random variable X . (d) Assume we have access to a uniform random variable U Uniform [ - 1 , 0] (Note the range of this variable). Suggest a function g ( . ) such that g ( U ) is a Cauchy random variable with parameter α .
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