elemprob-page19

# elemprob-page19 - Cauchy Here f(x = 1 1 1(x)2 What is...

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Cauchy . Here f ( x ) = 1 π 1 1 + ( x - θ ) 2 . What is interesting about the Cauchy is that it does not have ﬁnite mean, that is, E | X | = . Often it is important to be able to compute the density of Y = g ( X ). Let us give a couple of examples. If X is uniform on (0 , 1] and Y = - log X , then Y > 0. If x > 0, F Y ( x ) = P ( Y x ) = P ( - log X x ) = P (log X ≥ - x ) = P ( X e - x ) = 1 - P ( X e - x ) = 1 - F X ( e - x ) . Taking the derivative, f Y ( x ) = d dx F Y ( x ) = - f X ( e - x )( - e - x ) , using the chain rule. Since f X = 1, this gives f Y ( x ) = e - x , or Y is exponential with parameter 1. For another example, suppose X is N (0 , 1) and Y = X 2 . Then F Y ( x ) = P ( Y x ) = P ( X 2 x ) = P ( - x X x ) = P ( X x ) - P ( X ≤ - x ) = F X ( x ) - F X ( - x ) . Taking the derivative and using the chain rule, f Y ( x ) = d dx F Y ( x ) = f X ( x ) ± 1 2 x ² - f X ( - x ) ± - 1 2 x ² . Remembering that f X ( t ) = 1 2 π e - t 2 / 2 and doing some algebra, we end up with f Y ( x ) = 1 2 π x - 1 / 2 e - x/ 2 , which is a Gamma with parameters 1 2 and 1 2 . (This is also a χ 2 with one degree of freedom.) 10. Multivariate distributions.
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