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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I want to know the expectation of a product of independent beta distributed random variables.

I have an equation of the form $$Z = \frac{\prod_{i=1}^p X_i}{\prod_{i=1}^p Y_i}$$ where $$X_i \sim \mathcal{B}(\alpha_{x_i},\beta_{x_i})$$ and $$Y_i \sim \mathcal{B}(\alpha_{y_i},\beta_{y_i})$$ $X_i$ ...
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Beta and gamma function problem

Show that $$\int_0^1\frac{x^2}{\sqrt{1-x^4}}\,dx \times \int_0^1\frac{1}{\sqrt{1+x^4}}\,dx=\frac{\pi}{4\sqrt{2}}$$ I'm trying to solve this problem but I can't prove it. In the end, I got $\Gamma(3/8)...
Anish Bangia's user avatar
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Leading divergent term in integral before integration

Consider, for example, the Beta function: $$ B(\alpha,1+\alpha) = \frac{\Gamma(\alpha)\Gamma(1+\alpha)}{\Gamma(1+2\alpha)} = \int_0^1 dt\ t^\alpha (1-t)^{\alpha-1}. $$ Expanding the integrated result ...
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"How to solve a system of equations of estimators (alpha and beta) using the method of maximum likelihood (beta distribution) in R?"

...
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Dimension of maximum volumed unit ball

Let $V_n=V(B^n)$ be the volume of the $n$-dimensional unit ball $B^n$. By cross-sectioning $B^n$ along $x_n$-axis, $-1\leq x_n\leq 1$ and by means of similarity of hyper disks we have $$V_n=2\int_0^1(\...
Bob Dobbs's user avatar
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Gamma and Beta distributions [closed]

I am reviewing some statistics problems before my exam and I have stumbled upon a calculation that I am having trouble wrapping my head around. Could someone briefly explain to me how my professor ...
mathew's user avatar
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The conditional Expectation of the Beta Distribution

During a research project I was analyzing the shape of the conditional expectation of the Beta distribution, $g(t) = E[X\,|\, X>t]$ for $X\sim \mathrm{Beta}(\alpha,\beta)$. Using numeric ...
Matan Gibson's user avatar
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How to prove the relation between Beta and Gamma functions?

So, my teacher showed me this proof and unfortunately, it is vacation now. I don't understand what happened in the marked line. Can anyone please explain?
Plague's user avatar
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Vector version of integral that uses Beta function

I learned that a complex integral can be computed by using Beta function. I can compute $\begin{align*} \int_{\infty}^{\infty}(1+x^{2})^{-n}\mathrm{d}x. \end{align*}$ by using the following relation. $...
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Calculate the integral in the Beta function using Newton binomial theorem.

I want to prove the relationship between the Beta function and the Gamma function, i.e. $$ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}. $$ I want to start from expanding $(...
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What is the value of beta distribution at $x=0$, when $\alpha, \beta < 1$?

I am a bit confused as to how the beta distribution can be defined at $0$. The formulation of the beta distribution is given as - $f(x)=\frac{\Gamma(a+b)}{\Gamma(\alpha) \Gamma(\beta)} x^{\alpha-1}(1-...
Anirban Chakraborty's user avatar
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1 answer
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Integral related to $\frac{\sqrt{2}K(\sqrt{\frac{2k}{1+k}})}{\sqrt{1+k}}$.

Context: While playing with this integral: $$I(k)=\int_{0}^{\pi/2}\frac{\sqrt{1+\sqrt{1-k^2\sin^2x}}}{\sqrt{1-k^2\sin^2x}}dx\tag{1}.$$ Then I noticed: $$I(1/3)=\sqrt{3}\int_{0}^{\pi/2}\frac{\sqrt{3+\...
User's user avatar
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Sum of uniform and beta distribution

Suppose $X ∼ Beta(a = 3; b = 1; θ = 1)$ and $Y ∼ U (−2, 2)$ are independent. Derive an expression for the cumulative distribution function of $X + Y$. I am trying to do this by a convolution but I am ...
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Analytic continuation of incomplete beta function

The incomplete beta function $B_x(a,b)$ is defined for $x\in [0,1]$ by the integral $$B_x(a,b)=\int_0^x dt t^{a-1}(1-t)^{b-1}.\tag{1}$$ I'm interested in two aspects associated to that. First is the ...
Gold's user avatar
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Is Beta function essential to evaluating the integral $\int_0^{\infty} \frac{d x}{\left(1+x^4\right)^n}$?

After investigating the indefinite integral in the post, I found that $$\int \frac { d x } { x ^ { 4 } + 1 } = \frac { 1 } { 4 \sqrt { 2 } } \left[ 2 \tan ^ { - 1 } \left( \frac { x ^ { 2 } - 1 } { ...
Lai's user avatar
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How to find partial derivatives of the Beta Function?

I was reading the book (Almost) Impossible Integrals, Sums and Series. The author used a method involving taking partial derivatives of the Beta Function to solve some integrals. $$B(x,y)=\int_0^1u^{x-...
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Proof on representing the incomplete beta function as a hypergeometric function

The incomplete beta function is defined as $$B(x;c,d) = \int_0^xt^{c-1}(1-t)^{d-1}dt.$$ The hypergeometric representation of the incomplete Beta function is given by \begin{equation} B\left(x; c,d \...
Nate's user avatar
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Estimating factorial moments from a sample [closed]

I saw this article on wikipedia https://en.wikipedia.org/wiki/Beta-binomial_distribution and there is even a small example for the math-dummies like me. However, there is something I am not getting ...
Jonas Behr's user avatar
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regularized incomplete beta function integration

Solve $\int_{0}^{1}\frac{I_{u^{\frac{1}{p}}}\left ( p+\frac{1}{a} ,1-\frac{1}{a}\right )}{u}du$ . In Mathematica, this integral does not converge but from an article, I got the answer to this integral ...
ASHLIN VARKEY's user avatar
3 votes
1 answer
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Solve $\sum_{k=1}^\infty B(k,k)$

I have solved$$\sum_{k=1}^\infty B(k,k)$$Like so: $$\sum_{k=1}^\infty B(k,k)=\int_0^1\sum_{k=0}^\infty t^k(1-t)^kdt=\int_0^1\frac1{1-t+t^2}dt=\int_0^1\frac1{(t-\frac12)+\frac34}dt=\int_{-\frac12}^\...
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Is there a closed-form result of this integral with imcomplete beta function?

The integral is $\int_{\sqrt{|T^2-w^2|}}^{T+w} \left(1-I_{\sin^{2}\alpha}\left(\frac{n+1}{2},0.5\right)\right)^{M-1}x^{n-1}I_{\sin^{2}\beta}\left(\frac{n-1}{2},0.5\right)dx$ where $I$ is the ...
jobs adam's user avatar
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Mapping the upper half-plane onto rhombus

Book's question: Map the upper half-plane $\Im z>0$ onto a rhombus in the $w$-plane with angle $\alpha\pi$ at the vertex $A=0$ and side $d$. The correspondence of the points is given by $A=0\to z=0$...
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Expected value of the parameter $P$ of a Bernoulli having observed $s$ successes over $t$ trials, when $P$ is uniform on $[a,b]$ with $0\le a<b\le 1$

Using the relation between Beta and Gamma functions $$B(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$$ where $$B(m,n) = \int_{[0,1]} p^{m-1}(1-p)^{n-1}\operatorname{d}p$$ and the fact that for any $n ...
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Derivation of the cumulative distribution function for the beta-binomial distribution

Let $X$ be a random variable following a beta-binomial distribution: $$ X \sim \mathrm{BetBin}(n, \alpha, \beta) \; . $$ According to Wikipedia, the cumulative distribution function of $X$ is $$ F_X(x)...
Joram Soch's user avatar
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2 answers
158 views

Analytic continuation of Dirichlet beta funcion

In terms of Hurwitz zeta function, Dirichlet beta function is given by $$\beta(s)=\frac1{4^s}\left(\zeta(s,\frac14)+\zeta(s,\frac34)\right).$$ Following the links, by means of analytic continuation of ...
Bob Dobbs's user avatar
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2 votes
4 answers
153 views

How to find the limit $\left (\int_0^{\frac{\pi}{2}}\sin^n xdx\right)^{\frac{1}{n}}$

This is equivalent to $\left (\frac{(2n-1)!!}{2n!!}\right)^{\frac{1}{n}}$. I try to add some terms to the denominator, so as to use the Stirling formula. Then $$\left (\frac{((2n-1)!!)^2}{2n!}\right)^{...
Ychen's user avatar
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To evaluate some Complex integral by using Beta function

My book says that $$\int_{r-i\infty}^{r+i\infty} O(\vert s \vert^{-r}) \vert ds \vert = K\int_{0}^{\infty} \frac{1}{(1+t)^r}dt $$ ( where K, $r \space(r> 0)$ constants, s complex ) But I can't ...
David Lee's user avatar
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"Legendre-type" integrals involving $\frac{dt}{\sqrt{t^2-2t\cos(\theta)+1}}$

Summing Legendre polynomials $P_{l}(\cos\theta)$ often leads to expressions containing $\frac{1}{\sqrt{t^2-2t\cos\theta+1}}$, as this is the generating function for the Legendre polynomials. I want to ...
Luke's user avatar
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Solve for $\int_0^1 t^{s+a-1}(1-t)^{a-1}(2t-1)^{-1}dt$

Solve for $$\int_0^1 t^{s+a-1}(1-t)^{a-1}(2t-1)^{-1}dt$$ I first convert $(2t-1)^{-1}$ to its binomial and then geometric form following solve for (2t-1)^-1 in binomial and hypergeometric form $$-\...
joe_bill.dollar's user avatar
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Is there a way to express $ \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}\,\,B_{\frac{1}{2}}\left(n+\frac{1}{2},\frac{1}{2}\right)$

Trying to answer this question, I face the problem of making explicit (even in terms of special functions) $$ \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}\,\,B_{\frac{1}{2}}\left(n+\frac{1}{2},\frac{1}{2}\...
Claude Leibovici's user avatar
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Expansion of Beta function near negative integers

Is there a closed form expression for the coefficients (after Taylor expansion) of the Beta function $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ around an arbitrary negative integer let's say $x=-...
Hkw's user avatar
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Express the following $\int \cdots \int \prod_i AR_i ^{\alpha_i-s_i}dR_{k-1} \cdots dR_1$ in binomial form

I have the following integral series, $$\int \cdots \int \prod_i AR_i ^{\alpha_i-s_i}dR_{k-1} \cdots dR_1$$ Which is part of the beta theorem, and here $R_k$ stands for $1-R_1-\cdots-R_{k-1}$, so we ...
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Integration: "partial" Beta function?

It is well known that the Beta function is defined as $$B(x,y) =\int\limits_0^\infty \frac{t^{x-1}}{{(1+t)}^{x+y}}\,\mathrm{d}t$$ What if I have the exact same integral but with a non-zero lower bound,...
Da Li's user avatar
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Challenging integral evaluation

I computed the following integral and got that $$\int_S [x_1y_3+x_2(y_2+y_3)+x_3(y_1+y_2)]^n =O\Big(\frac{\log(n)}{n^3}\Big).$$ The integral is over the set $S=\{x_1+x_2+x_3=1, y_1+y_2+y_3=1 | x_i,y_i\...
Omer Moyal's user avatar
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Natural two parameter probability distribution for a one-sided interval

On the reals, a normal distribution with mean $\mu$ and standard deviation $\sigma$ could be considered somewhat natural (loose terminology, but a lot of stuff in nature occurs normally distributed, ...
Fraser's user avatar
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2 answers
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How to solve this definite integral? $\int_0^1\frac{1}{\sqrt{1+x^4}}\,{\rm d}x$

$$I=\int_0^1\frac{1}{\sqrt{1+x^4}}\,{\rm d}x$$ I am trying this question by substituting $$x=\sqrt{\tan(y)}$$ And finally arrived at a form of $$\sqrt{2\operatorname{csc}(2y)}$$ But how to proceed ...
Subha Sankar Roy's user avatar
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2 answers
61 views

How to compute $\int_{-\infty}^{\infty} 1/(x^2+1)^s dx$? [duplicate]

How does one evaluate $$\int_{-\infty}^{\infty} (x^2+1)^{-s} dx$$ in the domain $\operatorname{Re}(s) > \frac{1}{2}$? Specifically, I would like to show that $$\int_{-\infty}^{\infty} (x^2+1)^{-(s+...
D_S's user avatar
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Need help in detailing the proof of the existence of the test function

For every compact subset $K\subset \mathbb R^n$ and every $\epsilon>0$ $\exists$ a test function $\psi\in C_c^{\infty}(\mathbb R^n)$ such that (a) $0\le\psi(x)\le 1\forall x\in \mathbb R^n$ (b) $\...
Styles's user avatar
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1 vote
1 answer
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Evaluate the indefinite integral $\int\left(1-\left(1-x^{\frac{1}{a}}\right)^a\right)\mathrm dx\,$

I am currently working on the following expression: $$\int\left(1-\left(1-x^{\frac{1}{a}}\right)^a\right)\mathrm dx$$ I am particularly interested in finding the closed-form solution for this integral....
JuanmaGun's user avatar
4 votes
6 answers
135 views

Proving that $I=\int_0^\infty \frac{n x^{n-1}}{(1+x)^{n+1}}dx<+\infty$

I got this integral from a probability question in which I am interested to prove that it is $<\infty$ $$I=\int_0^\infty \frac{n x^{n-1}}{(1+x)^{n+1}}dx,~~~n \ge 1$$ I tried substitution $y =x+1$, ...
some_math_guy's user avatar
0 votes
2 answers
137 views

A pattern among the polynomial integrals $\int_\alpha^\beta(x-\alpha)^j (x-\beta)^k\,dx$

I've known, for $\beta > \alpha$, $\int_\alpha^\beta(x-\alpha)(x-\beta)dx=-\frac16(\beta-\alpha)^3$ for a while, and I've experimented with its "relatives" such as $$\int_\alpha^\beta(x-\...
sreysus's user avatar
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2 votes
2 answers
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On bodies of revolution for $y =(1-x^q)^p$

This question is posted in response to a recent one seeking the volume of $y =(a^{2/3}-x^{2/3})^{3/2}$ rotated about the x-axis. I wondered why people don't seek a more general solution when posed ...
Cye Waldman's user avatar
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Beta or Incomplete Beta Integral.

I have derived along the way and I have reached a situation as below. After further research on myself, I noticed it has a pattern as Beta/Incomplete Beta Function. Could anyone please give me some ...
Gambit's user avatar
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3 votes
1 answer
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Compute the integral $\int_0^\infty t^{3/\xi-1} e^{-t} \Gamma(\frac{2}{\xi},t) \ dt$

Hopefully this is my last question of this kind... I've tried this in several different ways, but I always end up stuck in a loop of iterative integration by parts and proving 1 = 1. As important a ...
FairyLiquid's user avatar
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1 answer
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Inverse of a gamma function

Assuming I have some relatively simple equation: $$ \eta(\alpha) = \frac{\Gamma\left(\frac{3}{\alpha}\right) \Gamma\left(8-\frac{3}{\alpha}\right)}{\Gamma\left(8\right)}, $$ is there any simple way to ...
FairyLiquid's user avatar
-1 votes
1 answer
83 views

Evaluate: $\int_{0}^{\infty} x^{a} (1+x)^{b} dx$ [closed]

I want to express \begin{align} \int_{0}^{\infty} x^{a} (1+x)^{b}~ dx \end{align} with Gamma functions. Recall that the beta function is defined as \begin{align} B(z_1, z_2) = \int_0^1 t^{z_1-1} (1-t)^...
phy_math's user avatar
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1 vote
1 answer
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On a tricky integral and Euler's reflection formula

I've been trying to compute the following integral $\forall s\in]0;1[$: $$I=\int_0^{\frac{\pi}{2}}\cot^{2s-1}(\theta)\ d\theta$$ The limits smelled a lot like the Beta function, and sure enough, ...
Jacques Tarr's user avatar
1 vote
3 answers
158 views

Evaluating $\displaystyle\int_{0}^{\infty} \frac{{x}^{2/3}-{x}^{1/4}}{(1+{x}^{2})\ln(x)} dx$ using the gamma and beta functions

How do I evaluate this integral using the gamma and beta functions? $$\int_{0}^{\infty} \frac{{x}^{2/3}-{x}^{1/4}}{(1+{x}^{2})\ln(x)} dx $$ I don't know what substitution I should make in order to use ...
Alex's user avatar
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4 votes
2 answers
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Generalisation of a definite integral $I(a)$ with value independent of its parameter $a$.

When I first met the integral $$I(a)=\int_0^{\infty} \frac{1}{\left(1+x^a\right)(1+x)^2}d x,$$ having the value $ \frac12$, which is decent and independent of $a$. Then I want to generalised it in a ...
Lai's user avatar
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Integrate without using the property that the function is odd

Integrate using substitution(not trigonometric) the following. I will share how I did it, even though the answer is not right. $$\int_{-1}^1 x \sqrt{1-x^2} dx $$ $$ Substituting \space x^2=y $$ $$...
Statistics aspirant's user avatar

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