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post14 - STAT 511-2 Spring 2012 Lecture 14 Jun Xie Finish...

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1 STAT 511-2 Spring 2012 Lecture 14 Feb 10, 2012 Jun Xie Finish the content of cumulative distribution function in the last post. 4.2 Expected values Definition The expected or mean value of a continuous rv X with pdf f ( x ) is x = E ( X ) = 𝑥𝑓 𝑥 𝑑𝑥 −∞ . Proposition If X is a continuous rv with pdf f ( x ) and h ( X ) is any function of X , then E [ h ( X )] = h ( X ) = ( 𝑥 ) 𝑓 𝑥 𝑑𝑥 −∞ . Example 11 Two species are competing in a region for control of a limited amount of a certain resource. Let X = the proportion of the resource controlled by species 1 and suppose X has pdf of uniform on [0,1]. Then the species that controls the majority of this resource controls the amount h ( X ) = max ( X , 1 X ). The expected amount controlled by the species having majority control is then E [ h ( X )] = 𝑚𝑎𝑥 ( 𝑥 , 1 − 𝑥 ) 𝑓 𝑥 𝑑𝑥 −∞ = 1 − 𝑥 𝑑𝑥 1/2 0 + 𝑥𝑑𝑥 1 1/2 = 1 2 1 2 𝑥 2 | 0 1/2 + 1 2 𝑥 2 | 1/2 1 = 3 4 . Definition The variance of a continuous random variable X with pdf f ( x ) and mean value is
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