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I don't know if this question is too trivial for you guys, it is not for me though...

Is there a closed-form equation for $b = \int_{0.5}^{1} Beta(\alpha,\beta)$?

$$ Beta = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} $$ $$ B(\alpha,\beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} $$

Google search doesn't help much when querying for a mathematical equation, I really tried...

DBS
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    What are you integrating, the density or the distribution, or something else? Note that the integral of the density is the regularized [incomplete beta function](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) which is not generally in closed form but might be doable in particular instances. – Glen_b Jun 09 '16 at 02:02
  • The PDF (Probability density function) – DBS Jun 09 '16 at 02:04
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    please edit the density into your question. What do you need this in closed form for? (It's easy enough to evaluate in practice). Do you really need the general case? – Glen_b Jun 09 '16 at 02:06
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    It is rather sloppy to omit the differential in this case. Which variable are you concerned with? – J. M. is not a statistician Jun 09 '16 at 07:18

1 Answers1

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There is ... but it is complicated, and not sure if that useful. I am assuming the integral you want is $$ \int_{0.5}^1 \text{beta}(x; \alpha, \beta) \; dx $$ where the beta density is $\text{beta}(x; \alpha,\beta)=\frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)}$.

With some help from maple this integral is evaluated as $$ {\frac {1}{{\rm B} \left(\alpha,\beta\right)\alpha} \left( - {\mbox{$_2$F$_1$}(\alpha,-\beta+1;\,1+\alpha;\,{\frac{1}{2}})}{2}^{- \alpha}+{\frac {\Gamma \left( 1+\alpha \right) \Gamma \left( \beta \right) }{\Gamma \left( \alpha+\beta \right) }} \right) } $$

kjetil b halvorsen
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    Doesn't this just reformulate the question? How would one go about computing values of $_2F_1$ accurately? (It's not simple to do so in full generality.) – whuber Oct 26 '21 at 19:35
  • Maybe --- depends on what is the purpose of the question – kjetil b halvorsen Oct 26 '21 at 19:42
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    Anyone who does not conceive of an regularized complete Beta function as a "closed form" is *a fortiori* not going to accept an algebraic combination of Gamma and hypergeometric values as closed! – whuber Oct 26 '21 at 19:46
  • Yes, that is *not* closed form in any sense I have seen the term used. – Ben Oct 26 '21 at 23:21