I have a variable X that follows the chi-square distribution with one degree of freedom. Is there anything known about the distribution of $e^X$?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Given $X \sim \chi^2(1)$, let $Y=e^X$. I do not know the name of the distribution of $Y$. But I believe the density of $Y$, denoted by $p(y)$, takes the following form. Let $f(x) = \frac{1}{\sqrt{2 \pi}} x^{-1/2} e^{-x/2}$ be the density of $X$.
\begin{align*} p(y) &= \frac{d x}{d y} f(x) \\ &= \frac{d \log(y)}{d y} f(\log{y}) \\ &= \frac{1}{\sqrt{2\pi} y^{3/2} \sqrt{\log{y}}}, \end{align*}
defined for $y>1$.
