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{\bf Question}

Suppose that $X$ is uniformly distributed over the interval (0,1),
i.e. the pdf of $X$ is

$$f(x)=\left\{ \begin{array} {ll} 1, & {\rm for}\ 0<x<1;\\ 0, &
{\rm otherwise} \end{array} \right.$$

Find the pdf of $Y=-2\log(X)$.


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{\bf Answer}

$\displaystyle y=-2 \log x \Rightarrow x=e^{-\frac{y}{2}}$

The transformation is decreasing.

The range of y is $0<y<\infty$.

$\displaystyle
\frac{dx}{dy}=e^{-\frac{y}{2}}\left(\frac{1}{-2}\right)$

The pdf of $Y$ is $\displaystyle
g(y)=1\left|\frac{dx}{dy}\right|=\frac{1}{2} e^{-\frac{y}{2}},\ \
\ 0 < y < \infty.$

The distribution is called the $\chi ^2$ distribution with 2
degrees of freedom.

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