4
$\begingroup$

I have a variable X that follows the chi-square distribution with one degree of freedom. Is there anything known about the distribution of $e^X$?

$\endgroup$
5
$\begingroup$

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 $y>1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.