Questions tagged [beta-function]

For questions about the Beta function, a special function closely related to the Gamma function. It is advisable to also use the [special-functions] tag in conjunction with this tag.

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1answer
694 views

Integral of the incomplete beta function

Does anyone know how to evaluate $$I(a,b) = \int_0^1 B_t(a,b) dt$$? Here $B_t(a,b)$ is the incomplete Beta function, defined as $$B_t(a,b) = \int_0^t x^{a-1}(1-x)^{b-1} dx $$ Of course $I(a,b)$ ...
2
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2answers
223 views

Using the Eulerian integrals evaluate $\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x$

The question asks to evaluate the integral: $$\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x.$$ I have tried a few substitutions but am not getting anywhere. Thanks in advance!
4
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1answer
565 views

Double integral involving beta functions

I want to prove the following result: $$\int_0^1 \int_0^1 {f(xy)(1-x)^{p-1}y^p(1-y)^{q-1}} \mathrm{d}x \, \mathrm{d}y=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)} \int_0^1 {f(t)(1-t)^{p+q-1}} \mathrm{d}t.$$...
3
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0answers
105 views

Evaluate $\int_0^1\frac{\ln{\left(1-kt^2(1-t)+\frac{t^4(1-t)^2}{4}\right)}}{t}dt$ , where $k=\cos1$

I want to find the value of $\displaystyle\sum_{n=1}^\infty \dfrac{\cos n}{n^22^n\binom{3n}{n}}.$ Since $\displaystyle\dfrac{1}{\binom{3n}{n}}=2n\beta(2n,n+1)=2n\int_0^1t^{2n-1}(1-t)^ndt$, $\...
3
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0answers
417 views

Given the parameters of a Beta Distribution, how do I calculate the probability of a specific value?

I'm down the rabbit hole on a least squares spline approximation problem I'm working on. I've solved the least squares piece, as well as calculating a statistic called "Q1" (the Durbin-Watson ...
0
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1answer
241 views

kth Moment of Beta Distribution

I was reading the following about the Beta Distribution: $E({ X }^{ n })=\frac { \beta (n+a,b) }{ \beta (a,b) } =\frac { \Gamma (n+a)\Gamma (b)\Gamma (a+b) }{ \Gamma (n+a+b)\Gamma (a)\Gamma (b) } =\...
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1answer
66 views

Proving that: $\int_0^{\infty}{\dfrac{\cosh(zyt)}{(\cosh(t))^{2x}}dt}=2^{2x-2}\cdot\dfrac{\Gamma(x+y)\Gamma(x-y)}{\Gamma(2x)}$

How can i prove that: $$\int_0^{\infty}{\dfrac{\cosh(zyt)}{(\cosh(t))^{2x}}dt}=2^{2x-2}\cdot\dfrac{\Gamma(x+y)\Gamma(x-y)}{\Gamma(2x)}$$ I only can see that: $$\dfrac{\Gamma(x+y)\Gamma(x-y)}{\Gamma(...
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0answers
136 views

Gamma Triplication formula using Beta Function

Recently i have seen that you can develope the Legendre Duplication Formula by using Beta FUnction. http://mathworld.wolfram.com/LegendreDuplicationFormula.html And i´m interesting to develope the ...
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1answer
67 views

finding the closed-form of $k$ in the series $\sum_{n=1}^{m}\beta (2n-1)\zeta (2n)=m+k$

finding the closed-form of $k$ in the series $$\sum_{n=1}^{m}\beta (2n-1)\zeta (2n)=m+k$$ when m go to infinity from some values of $m$, I found the $$k=0.358971008185307705...$$ any help, thanks
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0answers
50 views

$\lim_{\alpha\rightarrow 0}\left(\alpha^{(m+1)/m-2/n}\int_0^{\tan^{-1}(\alpha^{-1})}\sin^{1/m}(\theta)\cos^{2(1-n)n-1/m}(\theta)\:d\theta\right)=$?

Is there any way to evaluate the limit \begin{align}\lim_{\alpha\rightarrow 0}\left(\alpha^{(m+1)/m-2/n}\int_0^{\tan^{-1}(\alpha^{-1})}\sin^{1/m}(\theta)\cos^{2(1-n)n-1/m}\left(\theta\right)\:d\theta\...
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1answer
969 views

Beta distribution times a scalar

If I have a random variable that has a Beta distribution multiplied by a scalar (say 1000), what is its distribution then? I have been doing some research and it appears not to be a beta distribution. ...
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3answers
62 views

How to prove the summation $\sum\limits_{i=0}^b {b \choose i}(-1)^i\frac{1}{a+i+1}=\frac{a!b!}{(a+b+1)!}$ [closed]

I am working on the Beta function integration. After the integral, I want to prove the summation: $$\sum_{i=0}^b {b \choose i}(-1)^i\frac{1}{a+i+1}=\frac{a!b!}{(a+b+1)!}$$ Can anyone give an idea?
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0answers
346 views

How to derive Clopper-Pearson interval's F and beta approximation?

It is well-known that there is an approximation of the Clopper-Pearson exact Confident Interval for binomial test. Wiki It just simply claimed, without any reference that: Because of a relationship ...
6
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1answer
87 views

If $ I = \int_{0}^{1}\left[1-(1-x^2)^{100}\right]^{201} xdx$ and $J=\int_{0}^{1}\left[1-(1-x^2)^{100}\right]^{202}xdx\;,$ Then $ \frac{I}{J}$

If $\displaystyle I = \int_{0}^{1}\left[1-(1-x^2)^{100}\right]^{201}\cdot xdx$ and $J=\int_{0}^{1}\left[1-(1-x^2)^{100}\right]^{202}\cdot xdx\;,$ Then value of $\displaystyle \frac{I}{J}$ $\bf{My\; ...
10
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2answers
291 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where $a=+~\dfrac1n$...
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1answer
897 views

Beta Distribution and Expectation and Variance

Let X ∼ Beta(a, b). Compute E[X] and Var(X). So confused how to start this question. Any hint would be helpful.
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1answer
223 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
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1answer
52 views

Infrequent fail of the popular parameter estimators, having several beta-distributed random variables to be estimated

I have a project in which there exist $N$ Beta-distributed Random variables each of which should be estimated, having a sample for each of them. The sample domain is $\{0.1,0.3,0.5,0.7,0.9\}$ and the ...
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1answer
154 views

Beta-gamma integrals

I know that $B (m,n) =\frac{\Gamma (m) \Gamma (n)}{\Gamma(m+n)}$. But $B (m,m)$ = $2^{1-2m} B(m, 1/2) $, I didn't get how this equation arrived, I tried applying the formula but cant obtain all the ...
1
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1answer
459 views

Difference of Ordered Uniform Random Variables

Let $X_1, X_2,..., X_n$ be $n$ random variables distributed uniform(0,1) and $X_{(1)},X_{(2)},..., X_{(n)}$ be the ordered statistics of $X_1,...,X_n$ such that: $X_{(1)} < X_{(2)} < ... < ...
1
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1answer
169 views

Integral using gamma and beta functions

I can't solve this, no matter how I try $$\int_{-\infty}^{+\infty} \frac{e^{2x}}{4e^{3x}+9}\,dx$$ Thanks in advance
4
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2answers
151 views

Mistake with Integration with Beta, Gamma, Digamma Fuctions

Problem: Evaluate: $$I=\int_0^{\pi/2} \ln(\sin(x))\tan(x)dx$$ I tried to attempt it by using the Beta, Gamma and Digamma Functions. My approach was as follows: $$$$ Consider $$I(a,b)=\int_0^{\pi/...
11
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2answers
458 views

How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$

Evaluate $$\displaystyle \int^{\frac{\pi}{2}}_{0} x \ln(\cos x) \sqrt{\tan x}dx$$ Unfortunately, I have no idea on how to integrate this and thus cannot provide any inputs on my own. The only ...
0
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1answer
51 views

Beta function of the conjugate arguments

How to simplify (integrate) $$\text{B}\left(\dfrac{m}{2}+ix, \dfrac{m}{2}-ix\right)$$ when $m\in\mathbb{N}$? eg. when $m=1$ WolframAlpha simplifies the expression to $\pi \ \text{sech} (\pi x)$
2
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1answer
125 views

Rewrite Beta Functions in terms of Gamma Functions

Consider the following result in terms of Beta functions $B(\cdot, \cdot)$. $$ \mathbb E \left( \frac{Y_i^2}{\sigma^2} \right) = \frac{1}{r^2}\frac{n!}{(i-1)!(n-i)!} [B(n-i+1, i) -2B(n-i+r+1, i) + B(...
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1answer
2k views

Proof $\operatorname B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$

I'm trying to proof the equality $\operatorname B(x,y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all ...
2
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1answer
679 views

prove an identity involving beta function and gamma function

We know that $B(p,q)=\Gamma(p)\Gamma(q)/\Gamma(p+q)$ where $p, q>0$, and $B(p,q)$ is related to binomial coefficients if one of $p,q$ is an integer. I want to prove the following identity. $$\frac{...
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1answer
97 views

Derivation of continued fraction for the incomplete beta function?

Where can I find a derivation for this continued fraction representation of the incomplete beta function: http://dlmf.nist.gov/8.17#v? I would like to have a reference to the papers where this ...
1
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1answer
227 views

Formula for the Beta function for natural m, n

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function $B(x, y)$, it's symmetry $B(x,y) = B(y,x)$ aswell as the fact that $(x + y)B(x + 1, y) = xB(x, y) \space\...
9
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1answer
1k views

A difficult integral (expectation of the function of a random variable)

For $H>L$ , $p,q,\alpha,\beta>0$, and B(.,.) the beta functon, trying to solve this integral: $$\mathbb{E}(X)_0^H=\frac{\alpha H }{\beta B(p,q)}\int_0^H \frac{x \left(\frac{-H \log \left(\...
13
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1answer
276 views

$ \frac{\Gamma(r)\Gamma(s)\Gamma(k)}{\Gamma(r+s+k)} $ as a nice integral?

I saw the beta function: $$ \frac{\Gamma(r)\Gamma(s)}{\Gamma(r+s)}= \int_0^1 t^{(r-1)}(1-t)^{(s-1)} dt $$ and got me wondering if I could do something similar the product of 3 or more gamma ...
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0answers
89 views

What is $\int (1-e^{-x})^n dx$?

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
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0answers
21 views

Which beta distribution(s) has a variance `V` and a skew `S`?

Let X be a beta distributed random variable with parameters $\alpha$ and $\beta$, variance V and skew ...
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1answer
199 views

Bound on the Beta function

For positive integers x and y, we have that $$ B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} = \frac{1}{x} \left( \begin{array}{c} x+y-1 \\ x \end{array} \right)^{-1} . $$ However, $$ \left( \...
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2answers
197 views

Calculate improper integral using Euler's integral

I have to evaluate the following integral $$\int_0^2 \frac{dx}{\sqrt[5]{x^3(2-x)^2}}$$ Thanks in advance.
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1answer
75 views

Covariant and countervariant interpretations of SEQ in AR

Let a covariant interpretation of model $\mathscr{M}$ in model $\mathscr{M}'$ be defined as a couple of functions $(f,g)$, with $f$ injective, such that $$f:D\hookrightarrow\text{Seq}(\mathscr{M}')$$$$...
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1answer
3k views

Calculate integral using beta and gamma functions

I have to calculate the following integral using beta and gamma functions: $$ \int\limits_0^1 \frac{x\,dx}{(2-x)\cdot \sqrt[3]{x^2(1-x)}} $$ I came up with this terrible solution. Firstly, let's ...
6
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3answers
241 views

Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$

For the past week, I've been mulling over this Math.SE question. The question was just to prove convergence of $$\sum\limits_{n=1}^\infty\frac{n!}{\left(1+\sqrt{2}\right)\left(2+\sqrt{2}\right)\cdots\...
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4answers
84 views

How to show $I_p(a,b) = \sum_{j=a}^{a+b-1}{a+b-1 \choose j} p^j(1-p)^{a+b-1-j}$

Show that $$I_p(a,b) = \frac{1}{B(a,b)}\int_0^p u^{a-1}(1-u)^{b-1}~du\\= \sum_{j=a}^{a+b-1}{a+b-1 \choose j} p^j(1-p)^{a+b-1-j}$$ when $a,b$ are positive integers. I have no idea how to proceed. ...
2
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1answer
155 views

Integration with Beta Function $\beta$ [closed]

Given that: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty}\left({\sigma\,x^{-1}}\right)^u\beta\left(u,a\right)du=\left(1-{x \over \sigma}\right)^{a-1}$$ whereby $\sigma>0$, $a>0$ and $x$ is a real ...
4
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1answer
220 views

Prove that $\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$

Prove that the following integral: $$\int_0^{\infty} \int_0^{\infty} e^{-(x^2+y^2+2xy \cos \theta)} \,dx dy = \frac{\theta}{2\sin\theta}$$ The hints written on the book are beta function and to ...
2
votes
1answer
409 views

Integral over a simplex

Let $C_k$ be the $k$-simplex. I know that $$\int_{C_k} \prod_{i=1}^k x_i^{\alpha_i-1} dx_i = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^k \alpha_i\right)} \equiv B(\alpha_1,\ldots,\...
4
votes
2answers
7k views

Product of two Beta distributed random variables

I have two Beta distributed random variables : $X_1=B(\alpha_1, \beta_1)$ $X_2=B(\alpha_2, \beta_2)$ What can we say about $Y=X_1.X_2$? Is this also a Beta distributed random variable?
-2
votes
1answer
48 views

Beta density function

In this problem, I need to use Beta density function to solve the integration. $$ \int_{0}^{100}x^{2}\left(\,100 - x\,\right)^{2}\,{\rm d}x $$ After applying $\,{\rm Beta}\left(\, 3,3\,\right)$ I ...
-1
votes
1answer
73 views

Calculate $\int_{0}^t (0.3(1+x)^3+0.7)^{-1/2}\,dx$ [closed]

The following integral cannot be expressed in elementary terms: $\displaystyle\int_{0}^t \frac{1}{\sqrt{0.3(1+x)^3+0.7}}\,dx$ with $t>0$ real. What are possible techniques for approaching ...
1
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1answer
861 views

$E[X^2]$ of the Beta Distribution

So I know the Beta distribution is $$f(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\cdot x^{a-1}\cdot(1-x)^{b-1}$$ I know the $E[X^r] = \dfrac{\Gamma(a+b)\Gamma(a+r)}{\Gamma(a)\Gamma(a+r+b)}$ And I ...
10
votes
3answers
227 views

Harmonic number identity

I search for an elementary proof of the following identity: $$ \sum_{i=1}^{n-k} \frac{(-1)^{i+1}}{i}\binom{n}{i+k}=\binom{n}{k}\left(H_n-H_k\right) $$ I have found the following proof: $$ \sum_{i=1}^{...
2
votes
1answer
413 views

Why does β=α give a symmetric standard beta pdf?

I know that β=α is what will give a symmetric standard beta pdf, but why is this so?
3
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0answers
288 views

Definite integral similar to beta function but with exponential negative square root

I'm trying to solve the following definite integral: $\mathcal{I} = \int_0^1dx\ x^{P+k/2-m}(1-x)^me^{-\sqrt{x}}, $ where $P\in\mathcal{N}$ (whole positive numbers and zero), $m\in\mathcal{N}$, $k\in\...
4
votes
2answers
158 views

Log integrals II

By considering the integral \begin{align} I_{\mu} = \int_{0}^{\pi/4} \sin(2\theta) \, \left( \cos(\theta) - \sin(\theta) \right)^{\mu} \, d\theta \end{align} derivatives can be taken with respect to $...