Questions tagged [beta-function]

For questions about the Beta function (also known as Euler's integral of the first kind), which is important in calculus and analysis due to its close connection to the Gamma function. It is advisable to also use the [special-functions] tag in conjunction with this tag.

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Closed form of $\sum\limits_{m=0}^\infty \frac{(a)_m\text B_p(b+m,c-b)z^m}{m!}$ with the Incomplete Beta function.

This question comes from related “sum of incomplete gamma function” problems. Also,there are many research papers trying to make the Hypergeometric Incomplete. Here is one confluent example from ...
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Beta regression

Consider some positive random variables $X^1, X^2$ and $Y\sim Beta(p; 1)$ where $p=\beta_1X^1 + \beta_2X^2$. We have a random sample $\{X^1_i, X^2_i, Y_i\}$. Now, estimating $\beta_1, \beta_2$ is not ...
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Question on Rudin's exercise 8.21

Things to remark: $\Gamma(x) = \int_0^{\infty} t^{x-1}e^{-t}dt.$ $\int_0^1 t^{x-1}(1-t)^{y-1}dt$ = $\frac {\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ = $B(x,y)$. I don't understand from where do $(102)$ ...
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Baby rudin 8.21

Things to remark before 8.21. $\Gamma(x) = \int_0^{\infty} t^{x-1}e^{-t}dt.$ If $x>0$ and $y > 0$ , then $\int_0^1 t^{x-1}(1-t)^{y-1}dt$ = $\frac {\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ = $B(x,y)$. ...
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Derive Stirling's Approximation for Beta Function

I am trying to derive a Stirling's approximation for the Multivariate Beta function. This does not seem very hard, but I cannot even derive the Stirling's approximation for the usual Beta function. ...
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How Lebesgue dominated convergence theorem works when upper bound of an integral is not a constant? (Beta vs Gamma functions)

I am reading through the book Special Functions written by Roy, et al. At page 5, the limit is justified by Lebesgue dominated convergence theorem (LDCT). I didn't know this theorem so spent about a ...
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beta distribution as ratio gamma distributions

I need a proof of this statement please: Let $Y_1$ and $Y_2$ be independent random variables, where $Y_1$ is gamma distributed with parameters $\alpha$ and 1 and $Y_2$ is gamma distributed with ...
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Calculate $H= \int_0^\infty \frac{x^{n+1}}{1+x^n}dx, \, n>2$
Calculate the following integral: $H=\displaystyle \int \limits_0^\infty \dfrac{x^{n+1}}{1+x^n}dx, \, n>2$ I wanna use Beta function \$B(x,y)=\displaystyle \int \limits_0^\infty \frac{t^{x-1}}{(1+t)^...