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

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#### Best Answer

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{2pi} y^{3/2} sqrt{log{y}}}, end{align*}

defined for $y>1$.

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