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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 $\nu_\lambda$:

  1. It lies somewhere between a $\chi_3$ distribution (at $\lambda = 0$) and a Gaussian distribution (at $\lambda = \infty$ after some normalization)
  2. 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!)
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Please don't ask identical questions here and on math.SE, it leads to duplication of effort. math.stackexchange.com/questions/340205/… –  Dan Petersen Mar 25 '13 at 1:13
    
Thank you for telling me this! I did not know that that was a rule, I do now. I have deleted the one on math.stackexchange. Thanks again. –  Mihai Nica Mar 27 '13 at 3:31

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