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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.

3
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1answer
195 views

Recursive relationship for incomplete beta function

Consider the incomplete Beta function $I_x(a, b)$ $$ I_x(a, b) = \dfrac{B(x; a, b)}{B(a, b)} = \dfrac{\int_0^x t^{a-1} \left( 1-t \right)^{b-1} dt}{\int_0^1 t^{a-1} \left( 1-t \right)^{b-1} dt}. $$ ...
0
votes
1answer
72 views

hypergeometric function with arguments (1/3,1/2;3/2;x^6)

I am attempting to find an exact, closed, form of the integral: $\int_0^r (1-\frac{1}{r'^6})^{-\frac{4}{3}}dr'$ Wolfram evaluates this to: $=\frac{x\left((1-x^6)^{\frac{1}{3}} \;_2F_1(\frac{1}{3},\...
0
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0answers
91 views

Incomplete beta function and implicit function

Say $\,y\,$ is implicitly given by $$ I_y(\alpha+1,\beta) = \frac{1}{B(\alpha+1,\beta)}\int_0^y t^{\alpha}(1-t)^{\beta-1}dt=k\,. $$ where $\,0<k<1\,$ is a constant. Using Matlab for numerical ...
1
vote
1answer
199 views

Integral as a Beta function

Is there a way to express integrals like $$\int_0^1 x^{-a} (1+cx)^{1-a} \text dx$$ as a Beta function, where $c$ is positive (or negative) constant and $a>1$? I tried to use the substitution \...
0
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2answers
537 views

Limiting value of Beta function

I would like to know the limiting value of the Beta function: $$ B(a,b) = \int_0^1 x^{a -1}(1-x)^{b-1}\,d x~ \tag 1. $$ For instance, what does (1) reduce to as $ a \to \infty$ $ b\to \infty$ $ a\...
0
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1answer
122 views

A Beta function related integral

The integral $$\int \limits_{0} ^{1} \left( \frac{x}{x^2+a^2} \right)^p \frac{\log (x)}{x} dx$$ is being considered. Does anyone have an idea how to express it in terms of the Beta function? The ...
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2answers
183 views

Moments of beta distribution

I have a random variable given by $$ Y = a \cdot X \;, $$ where $X$ follows a beta distribution and $a$ is a simple constant. I want to find the moments of $Y$. I am aware of the general formula for ...
3
votes
1answer
680 views

Beta function series expansion

I have been trying to determine the series expansion of the beta function, but so far I haven't been successful. The two results I wish to obtain are the following: $$ B(x,y) = \sum_{n=0}^{\infty} \...
2
votes
2answers
409 views

Approximate distribution for sample mean of a small sample

Problem: Let $X_1,...,X_n$ be a random sample from a distribution with the density function $$f(x)=6x(1-x), \quad if \space 0\lt x\lt 1 \quad and\space 0 \space elsewhere$$ $$\bar{X}_n=\...
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2answers
4k views

Expected value of the k-th order statistic from uniform random variables

I am trying to find the expected value of $X_{(k)}$ Here is my work so far: $$f_{X_{(k)}}(x)=\frac{n!}{(k-1)!(n-k)!}f_X(x)[F_{X}(x)]^{k-1}[1-F_X(x)]^{n-k}$$ by $X_i \sim U(0,1)$ this becomes $$E(X_{(...
3
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2answers
525 views

Express this Integral in terms of the Beta Function

NOTE: The 'correct' solution in this post is actually incorrect; see Cye Waldman's correct solution. I am asked to solve the following integral for $E_n$, which appears in the study of the quartic ...
1
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2answers
187 views

$\int_0^{\frac{\pi }{2}} \frac{\sin ^{2 m-1}(\theta ) \cos ^{2 n-1}(\theta )}{\left(a \sin ^2(\theta )+b \cos ^2(\theta )\right)^{m+n}} \, d\theta$

I tried substituting $x = \sin^2{\theta}$ then $dx = 2 \sin{\theta} \cos{\theta}$ Which got me to: $\frac{1}{2}\int_0^1 \frac{x^{m-1} (1-x)^{n-1}}{ (a x+b (1-x))^{m+n}} \, dx$ But I have no idea ...
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0answers
188 views

Beta distribution: Sum of parameters

I use the beta distribution with bounds 0,1 as an approximation for the distributions of answers in a survey. Do you know wether distributions who have different alpha/beta ratios but the same sum ...
4
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0answers
134 views

Real methods for the evaluating $\int^{\pi/2}_{0}\cos(nt)\cos^m(t)\,dt$

One can show by integrating the following function $$f(z) = z^{n-m-1}(1+z^2)^m$$ around the following contour By equating the circular part of $|z|=1$ and the line on the imaginary part. $$\int^...
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3answers
88 views

Need help calculating a definite integral

I'm trying to calculate the following integral: $$ \int_0^1 e^{-\lambda(1-x)} (1-x)^{n-1} x^{k-n} dx $$ It seems like kind of a combination of Gamma and Beta function. I'm suspecting that it has ...
4
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0answers
326 views

Maximum of independent beta random variables.

Question: Fix a real number $\beta \geq 1$ and an integer $n \geq 1$. Let $X_1, \cdots, X_n$ be independent samples from Beta$(\beta,\beta)$. What is $$f(n,\beta) := \mathbb{E}\left[ \max\{X_1,\cdots,...
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2answers
352 views

Beta Function proof help

I'm trying to prove that beta(x,y)=beta(x+1,y)+beta(x,y+1) but not sure where to start. Any help would be greatly appreciated! Thanks
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0answers
49 views

By using the Beta-Gamma functions relation and the semigroup property of the volterra fractional integral operator

Can any one help me to show what did we do $$ \frac{\mathit{\lambda}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}\mathop{\int}\limits_{t0}\limits^{t}{{\mathrm{(}}{t}\mathrm{{-}}{s}{\mathrm{)}}^{{q}\...
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1answer
57 views

Proving that $B(2\alpha,1-\alpha)\le\frac1{\alpha(1-\alpha)}$

Let $0<\alpha<\frac12$, I cannot prove that $$ B(2\alpha,1-\alpha)\le\frac1{\alpha(1-\alpha)} $$ where $$ B(p,q):=\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)} $$ is ...
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0answers
232 views

Prove the following : $ \int_0^{ 1 } \frac{x^{l-1} (1-x)^{m-1}}{ (b+cx)^{l+m} }\,dx = \frac{\beta(l,m)}{(b+c)^{l}b^m} $

Prove the following : $$ \int_0^{ 1 } \frac{x^{l-1} (1-x)^{m-1}}{ (b+cx)^{l+m} }\,dx = \frac{\beta(l,m)}{(b+c)^{l}b^m} $$ I found this substitution : $$ x = \frac{-b}{t+c} $$ Using this, we get the ...
1
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1answer
682 views

Beta function with one negative and one positive arguments

Beta function defined as $B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$ is only well defined when Re $x,$ Re $y >0$. However, according to "http://www.efunda.com/math/beta/," we can use the fact that ...
3
votes
2answers
739 views

Evaluate: $\int_0^1 \frac{\left(1-x^4\right)^{3/4}}{\left(x^4+1\right)^2} \, dx$

Here is what I did: $\int_0^1 \frac{\left(1-x^4\right)^{3/4}}{\left(x^4+1\right)^2} \, dx$ Put $x^2=\tan (\theta )$ Then: $x=\sqrt{\tan (\theta )}$ and $dx=\frac{1}{2} \tan ^{-\frac{1}{2}}(\theta ) ...
3
votes
2answers
172 views

Beta function-like integral

$$\int\limits_0^1 \frac{x^{1-\alpha} (1-x)^\alpha}{(1+x)^3} \, dx $$ After the substitution $z=\frac{1}{x} - 1$, I've got this: $$\int\limits_0^\infty \left(\frac{1}{z}-1\right)^\alpha\left(\frac{1}...
0
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1answer
51 views

How to find the following limit $\lim_{x\to+\infty}x^{\alpha} B(\alpha, x), \hspace{20pt} \alpha > 0 $ [closed]

How to find the following limit $$\lim_{x\to+\infty}x^{\alpha} B(\alpha, x), \hspace{20pt} \alpha > 0, \alpha, x \in \mathbb{R} $$
0
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1answer
209 views

How to solve this integral using special functions

I came across an integral of this form: $$\int_{0}^{a}\frac{dx}{(a^{n}-x^{n})^{1/n}}$$ How do I solve this integral? I tried using this substitution: $x=asin^{2}(x)$ in order to reduce this to a beta ...
2
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1answer
499 views

Proving Beta prior distribution is conjugate to a negative binomial likelihood [closed]

Show that the Beta prior is conjugate to a negative binomial likelihood, i.e., if $\mathbf{X} | \theta \sim \mathrm{NegBin}(k,\theta)$ and $\theta \sim \text{Beta}(a, b)$, then $\theta | \mathbf{X} \...
2
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1answer
71 views

Proving that $ \pi = 2+\frac23 \sum_{n=1}^\infty \prod_{k=1}^{n-1} \frac{4k^2}{(2k+3)(2k+1)} $

In the course of trying to answer Closed form of a series involving the squared Beta function I came up with this expresion, which can be proven in a round-about manner via the Beta function: $$ \pi = ...
1
vote
2answers
116 views

Closed form of a series involving the squared Beta function

Is there a closed form for the expression $$\sum_{n=1}^{\infty}\left(n+\frac12\right)B^2\left(n,\frac32\right)$$ where $B(x,y)$ is the Beta function? Here is what I've done. I wrote: $$\sum_{n=1}^{\...
0
votes
1answer
28 views

analytical form of beta function, what is wrong about this?

This has to be a really stupid question, but here it goes: $$ \int_0^1 t^{x-1} (1 -t)^{y-1} dt= \int_0^1 t^{x-1} dt- \int_0^1 t^{x+y-2} dt= $$ $$ \frac{t^x}{x} |^1_0 + \frac{t^{x+y-1}}{x+y-1} |^1_0 =...
6
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1answer
154 views

Closed form expression of $\int_{-\infty}^{+\infty}dx \exp[-\alpha(x^2-a^2)^2]$

Is the following integral $$I(a,\alpha)=\int\limits_{-\infty}^{+\infty}dx \exp[-\alpha(x^2-a^2)^2]$$ analytically solvable i.e., have a closed form expression? Here, $\alpha, a$ are real positive ...
1
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2answers
102 views

Find the variance of a quadratic cost function

Given a beta distribution $f (y)=2 (1-y)$ and $C=10+20Y+4Y^2$, find $\mbox{Var} [C]$. I already know $\alpha=1$ and $\beta=2$, so $\mu =\frac13$ and $\sigma^2=\frac{1}{18}$. I also found that $\...
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0answers
300 views

How to generalize Reshetnikov's $\arg\,B\left(\frac{-5+8\,\sqrt{-11}}{27};\,\frac12,\frac13\right)=\frac\pi3$?

We have, $$\arg z_1 = \frac{k\,\pi}3, \quad z_1 = \left(\tfrac{1+\sqrt{-3}}{2}\right)^k\tag1$$ $$\arg z_2=\frac{k\,\pi}3, \quad z_2 = \left( B\Big(\color{blue}{\tfrac{-5+8\,\sqrt{-11}}{27}};\,\tfrac12,...
3
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0answers
113 views

On the integral $\int_0^1\frac{dx}{\sqrt[4]x\ \sqrt[4]{1-x}\ \sqrt{1-x\,\beta^2}}=\frac{2\pi}{7\sqrt{2}\,\beta}$ and $\cos\frac{2\pi}{7}$

(This continues the post on integrals that use roots of reciprocal polynomials.) Given $N=7$. First, how do we show that the algebraic number $\beta$ that solves, $$\int_0^1\frac{dx}{\sqrt[4]x\ \...
5
votes
1answer
279 views

A nice pattern for the regularized beta function $I(\alpha^2,\frac{1}{4},\frac{1}{2})=\frac{1}{2^n}\ $?

In this post, the problem was given integer/rational $N$, to solve for algebraic number $z$ in the equation, $$\begin{aligned}\frac{1}{N} &=I\left(z^2;\ a,b\right)\\[1.5mm] &= \frac{B\left(z^2;...
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0answers
283 views

Why do these two integrals use roots of reciprocal polynomials?

There is the nice integral by V. Reshetnikov, $$\int_0^1\frac{dx}{\sqrt[3]x\ \sqrt[6]{1-x}\ \sqrt{1-x\,\alpha^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{3}\;\alpha}\tag1$$ also discussed in this post. By ...
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0answers
75 views

Need help with an integral related to beta function

I've been struggling with a definite integral as below: $$ \int_0^1 x^{a-1}(1-x)^{b}(I_x(a+1,b) - 1) dx $$ where $I_{x}$ represents the regularized incomplete beta function. I'd initially tackled ...
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2answers
35 views

Substituting into an integral to create a Beta function

My probability textbook has a solution that turns a certain density function into something more familiar. Starting with, $\int_{-\infty}^{\infty} \frac{dx}{(1+x^2)^m}$ We substitute in: $v = (1 + ...
2
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4answers
108 views

A slight modification of Beta function

The Beta function is defined as follows: for real values $a,b>0$ $$B(a,b) = \int_0^1 w^{a-1}(1-w)^{b-1} dw.$$ It is known that $$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},$$ where $\Gamma$ ...
4
votes
1answer
78 views

Inequality for the Gamma function?

Let $x,y,a,b \geq 1$. I have the feeling that the following inequality is true: $$B(x+a,y+b)=\frac{\Gamma(x+a)\Gamma (y+b)}{\Gamma(x+y+a+b)}\leq \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}=B(x,y),$$ where $...
1
vote
2answers
184 views

Beta function with different integral limits

The beta function is $$B(x,y) = 2\int_{0}^{\frac{\pi}{2}} \cos^{2x-1}\theta \sin^{2y-1}\theta d\theta$$ for positive values of $x$ and $y$. How can this integral be used to evaluate other limits of ...
1
vote
2answers
676 views

Pdf of y =-logx when x is beta distributed

I want find the PDF of $Y$ when $Y=-\log X$ and $X$ has a beta distribution. I found the below formula as the answer, but I think there should be $(1-e^{-y})^{b-1}$ part should added to this. Is ...
1
vote
1answer
2k views

Integration, trigonometry, gamma/beta functions [closed]

In a classical mechanics text, I saw an integration(as part of a math) that uses gamma function as follows: $$\int_0^{\pi/2} (\cos^2 x-\cos^3 x)dx$$ $$=\frac{\Gamma(\frac{2+1}{2})\Gamma(\frac{0+1}{2})...
2
votes
0answers
215 views

Integral of product of two inverse regularized incomplete beta functions

I want to compute the integral of the product of two inverse regularized incomplete beta functions over $[0,1]$ in closed form, that is, to evaluate $$ J = \int_0^1 I_t^{-1}(a_1,b_1) \: I_t^{-1}(a_2,...
2
votes
1answer
147 views

Multiple integral over two dependent beta distributions

I want to evaluate this multiple integral: $$ \iiint\limits_{ \sum_{i=1}^4 x_i=1,\ \ \ x_1,\, x_2,\,x_3,\, x_4\, \ge \,0 } (x_1+x_2)^{N1} (x_3+x_4)^{N2} (x_1+x_3)^{N3} (x_2+x_4)^{N4} \, dX$$ For ...
2
votes
1answer
824 views

Posterior beta from prior beta distribution (features binomial)

We have the following trial - group of people and we count males. A person is male with probability $\theta$. Now, we need to estimate this parameter using additional data. As a start, we have a beta ...
0
votes
1answer
150 views

Creating a prior function for Bayesian interference of a Bernoulli

I am trying to solve exercise 3.12 from Kevin P.Murphy-Machine Learning_ A Probabilistic Perspective According to the question for a Bernoulli estimation the prior is $$ p(\theta) = 0.5 \ => \ ...
1
vote
2answers
429 views

Duplication formula for beta function

We use the beta function $\text B(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}\text dt$ to prove the duplication formula for the gamma function: $$\Gamma(2\alpha) = \frac {2^{2\alpha -1}\Gamma\...
2
votes
2answers
62 views

Find $c$ such that $cy^3(1-y)^8$ for $0 \leq y \leq 1$ is a PDF

I have a function $$f(y) = \left\{\begin{array}{cc} cy^3(1-y)^8 & 0 \leq y \leq 1 \\ 0 & \text{ Otherwise.}\end{array}\right. .$$ I want to figure out what ...
1
vote
1answer
603 views

A limit involving the Regularized Incomplete Beta Function

I'm trying to evaluate $$\lim_{n\to\infty} nx^{n-1}I\left(1-\frac{x^2}{4}; \frac{n+1}{2}, \frac{1}{2}\right)$$ where $I(x; a, b)$ is the Regularized Incomplete Beta Function and $-2\leq x \leq 2$, ...
-1
votes
1answer
86 views

Trying to Find Closed Form for Beta Integral

$$ \mbox{Consider}\quad\int_{0}^{1/2} v^{-t}\,\left(\, 1 - v\, \right)^{-\,\left(\, t + 1\, \right)} \,\,\,\mathrm{d}v $$ I have verified numerically that the integral converges for $t < 1$, but I ...