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In working in a probability application, I came upon a discrete random variable with support on $\mathbb N$ having p.m.f. $$\mathbb P(X=n)=\frac1n\binom{2(n-1)}{n-1}(1-p)^{n-1}p^n.$$ Here $p\in(0,1)$. I was wondering if this belongs to some parametric family, and if so, what is the name for that family?

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    $\begingroup$ This issue involving Catalan numbers is well treated here. See here as well $\endgroup$
    – Jean Marie
    Commented Mar 8, 2023 at 14:34
  • $\begingroup$ Thank you, those references are helpful! $\endgroup$ Commented Mar 9, 2023 at 10:13

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Some people call it Sibuya distribution.

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    $\begingroup$ This is thin enough at the moment that it could probably be a comment. It would be helpful to provide a source for this terminology. $\endgroup$ Commented Mar 8, 2023 at 21:45
  • $\begingroup$ I cannot find a Sibuya or Sibuya like distribution in the literature that seems to have the form as in my question. Do you have any reference? $\endgroup$ Commented Mar 9, 2023 at 9:26
  • $\begingroup$ Indeed, in this document by Sibuya himself, I don't find anything that is like the distribution in question. $\endgroup$
    – Jean Marie
    Commented Mar 9, 2023 at 16:28

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