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I have a distribution $Y \sim 1/(1-Beta(\alpha,\beta))$. I would like to understand its properties.

I was able to write down its density function, but I was not able to integrate it to get its mean, for example (except through simulation).

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The Beta is mirror-symmetric $1-\mathrm{Beta}(\alpha, \beta) =\mathrm{Beta}(\beta, \alpha)$, so you are essentially looking for the distribution of the reciprocal of a Beta random variable.

Also, if $X\sim \mathrm{Beta}(\beta, \alpha)$, then $1/X-1\sim\mathrm{BetaPrime}(\alpha, \beta)$, where $\mathrm{BetaPrime}$ is the Beta prime distribution.

So, I would say that your $Y-1$ is distributed $\mathrm{BetaPrime}(\alpha, \beta)$.

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    $\begingroup$ + I should have seen the symmetry. I still don't follow your BetaPrime argument (certainly due to my rusty algebra), but here is discussed the inverse of a Beta, which tells me most of what I need. Thanks. $\endgroup$ Nov 19 '12 at 15:56
  • $\begingroup$ Now I get the BetaPrime. Thx. $\endgroup$ Nov 19 '12 at 16:30
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    $\begingroup$ Your answer allowed me to put the end on this answer. $\endgroup$ Nov 27 '12 at 19:44

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