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If $X$ has the density function

f_\vartheta (x) = \Big \{ \begin{array}{cc} (\vartheta - 1)x^{-\vartheta} & x \geq 1\\ 0 & otherwise \end{array}

How can I see that $\log X \sim Exp(\vartheta - 1)$? I had the idea to look at $f(\log x)$ but I think that's not right. Many thanks for your help!

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    @t.b. There is a smoke signal $f$or you. And a cave painting. Not sure you get pinged i$f$ I send you a comment but I'm going to try.2014-06-18

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One way is to use the cdf technique. Let $Y = \log X$. Then $P(Y \leq y) = P(\log X \leq y) = P(X \leq e^y) = \int_1^{e^y} (\vartheta - 1)x^{-\vartheta} dx = \left.-x^{-\vartheta+1}\right|_1^{e^y} = 1-e^{-(\vartheta-1)y}.$ Differentiating with respect to $y$ yields $(\vartheta-1) e^{-(\vartheta-1)y}$ as the pdf of $Y$, which is also the pdf of an $Exp(\vartheta - 1)$ random variable.

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    Thanks for your very interesting answer. Students did not manage to have this effect on me. Yet... :-)2011-02-20