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In some work I was doing I derived a probability distribution that I do not recognize. Is it a known distribution?

$\Pr(X\le x)=\exp\left[-\frac{1}{2}\left(\frac{1}{2}x-\sqrt{1+\frac{x^{2}}{4}}\right)^{2}\right]$

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  • $\begingroup$ That square root reminds me of the Wigner semicircle distribution. You might try seeing if there is some nice transformation that takes you to it. $\endgroup$ Jan 11, 2012 at 14:45
  • $\begingroup$ The square root in my distribution comes from a hyperbola. The x/2 term flattens the positive-x asymptote against the x-axis, and the exponential maps 0 (well, $-\delta$) to 1 (or rather, $1-\epsilon$). $\endgroup$
    – Cyan
    Jan 11, 2012 at 15:41
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    $\begingroup$ Can I assume you've already looked through the Johnson, Kotz & Balakrishnan volumes? $\endgroup$
    – cardinal
    Jan 12, 2012 at 23:31

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