This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the following density:
$$ d\nu_\lambda(x)= \frac{4\sqrt{2}}{\lambda \sqrt{\pi}} e^{-\frac{1}{2} \lambda^2} \sinh (2 \lambda x) x e^{-2x^{2}} dx \text{ for } x>0 $$
This arises in a paper I am working on and was curious if it is a well known distribution.
Other facts about $\nu_\lambda$:
- It lies somewhere between a $\chi_3$ distribution (at $\lambda = 0$) and a Gaussian distribution (at $\lambda = \infty$ after some normalization)
- It arises as the distribution of $\max_{0\leq s\leq 1} B^{\lambda}(s)- \frac{1}{2}B^\lambda(1)$ where $B^{\lambda}$ is a Brownian motion with drift $\lambda$. (Drift $-\lambda$ would work too!)
