elemprob-fall2010-page32

elemprob-fall2010-page32 - e-x ) = 1-F X ( e-x ) . Taking...

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An exponential is the time for something to occur. A gamma is the time for t events to occur. A gamma with parameters 1 2 and n 2 is known as a χ 2 n , a chi-squared random variable with n degrees of freedom. Gammas and chi- squared’s come up frequently in statistics. Another distribution that arises in statistics is the beta: f ( x ) = 1 B ( a,b ) x a - 1 (1 - x ) b - 1 , 0 < x < 1 , where B ( a,b ) = R 1 0 x a - 1 (1 - x ) b - 1 . 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
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Unformatted text preview: 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 -f X (- x ) -1 2 x . 32...
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