# Tag Info

## Hot answers tagged beta-function

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### Show that $\sum_{n=0}^{\infty}\frac{2^n(5n^5+5n^4+5n^3+5n^2-9n+9)}{(2n+1)(2n+2)(2n+3){2n\choose n}}=\frac{9\pi^2}{8}$

Here is an answer based upon the arcsine function. We start with the following formula valid for $u\in(0,2)$ \begin{align*} \sum_{n=0}^\infty&\frac{2^{n-1}}{(2n+1)(2n+3)\binom{2n}{n}}u^n\\ &...
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### Writing the Beta Function in terms of the Gamma Function

$$f(\alpha, \beta, t) = \int_0^t x^{\alpha -1} (t-x)^{\beta -1}\;dx = \big(t^{\alpha - 1} * t^{\beta -1}\big)(t)$$ so using the fact that convolution is multiplicative in the Laplace domain, \begin{...
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### Is Beta function essential to evaluating the integral $\int_0^{\infty} \frac{d x}{\left(1+x^4\right)^n}$?

Any standard method (including scaling $x$ in the first display equation in the question statement) yields $$\int_0^\infty \frac{dx}{a + x^4} = \frac{\pi}{2 \sqrt 2} a^{-3 / 4} .$$ If $n$ is a ...
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### Compute $\int_0^1x^m(1-x^n)^pdx$

Hint. As @Did has noticed, one may recall the Euler Beta integral result: $$B(a,b)=\int\limits_0^1 t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}, \quad a>0,\,b>0.$$ ...
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### Integrating $\int_{0}^{1} x^a (c-x)^b dx$

Enforcing $x =ct$ gives $$\int_{0}^{1} x^a (c-x)^b dx= c^{a+b+1}\int_{0}^{c} t^a (1-t)^b dt =\color{red}{c^{a+b+1} \mathrm {B} (c;\,a,b) }$$ where $\mathrm {B} (c;\,a,b)$is the incomplete beta ...
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### How to compute $\int_0^1x^a(1-x)^be^{cx}dx$?

This is essentially the the moment generating function of the beta distribution. The result is hypergeometric and cannot be further simplified.
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### Expected value of the k-th order statistic from uniform random variables

I think you're messing up in this way: $\Gamma(m) = (m-1)$! for a positive integer $m$. I'm getting \begin{align*} E(X_{(k)}) &=\frac{n!}{(k-1)!(n-k)!}\int_0^1 x^{k}[1-x]^{n-k}dx \\[5pt] &= ...
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