02_23ans - STAT 400 Examples for Spring 2011 1 Let X be a...

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STAT 400 Examples for 02/23/2011 Spring 2011 1. Let X be a continuous random variable with the probability density function f ( x ) = k x , 0 x 4, f ( x ) = 0, otherwise. a) What must the value of k be so that f ( x ) is a probability density function? 1) f ( x ) 0, 2) ( ) 1 d = - x x f . ( ) = = = - 4 0 2 1 4 0 d d d 1 x x k x x k x x f = = 3 16 0 4 3 2 2 3 k x k . k = 16 3 = 0.1875 . b) Find the cumulative distribution function of X, F X ( x ) = P ( X x ). F X ( x ) = P ( X x ) = ( ) - x d y y f . x 0 F X ( x ) = 0. 0 x 4 F X ( x ) = 2 3 2 3 0 8 1 0 3 2 16 3 16 3 x x y y y x d = = . x 4 F X ( x ) = 1.

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c) Find the probability P ( 1 X 2 ). P ( 1 X 2 ) = ( ) 2 1 d x x f = 1 2 3 2 16 3 d 16 3 2 3 2 1 x x x = 0.22855 . OR
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This note was uploaded on 02/24/2011 for the course STAT 400 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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02_23ans - STAT 400 Examples for Spring 2011 1 Let X be a...

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