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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.

10
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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,...
5
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0answers
109 views

Evaluate $\int_0^1 \log^n(x^a)\log^m(1-x^{\color{red}{\alpha}})x^b(1-x^{\color{red}{\beta}})^t\mathrm dx$ with $\alpha\ne\beta$

Recently dealing with algebraic integrals composited of logarithms and polynomials. I learned about using the derivatives of the Beta Function in order to evaluate them. Applying this knowledge I was ...
5
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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 ...
4
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52 views

Finding an elementary evaluation of $B_{1/2}(a,1-a)$

I'm trying to prove $$B_{1/2}(a,1-a):=\int_0^{1/2}x^{a-1}(1-x)^{-a}dx=\int_0^1\frac{x^{a-1}-x^a}{1-x^2}dx$$ $(a>0)$ (where $B$ denotes the incomplete beta function) with elementary method. I have ...
4
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0answers
87 views

Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular.

Let $A$ be an $n\times n$ matrix whose entries are \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where $0 < \lambda_{1} < \ldots < \lambda_{n}$ and $0 < \mu_{1} < \...
4
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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^...
4
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327 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,...
4
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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 ...
3
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88 views

Why does $\int_0^1 \frac{\ln(\ln(p))}{1+p^2}dp$ Converge?

I was messing around with the Dirichlet Beta Function and was able to get a formula: $$\int_0^1 \frac{\ln(\ln(p))\ln(p)^{x-1}}{1+p^2}dp = \Gamma(x)(-1)^x(\beta'(x)-\beta(x)(i\pi +\psi(x)) $$ where $\...
3
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0answers
137 views

Closed form for $\int\limits_0^\infty \frac{x^\alpha(1-x)^\beta}{(x-c)^\gamma(x-b)^\gamma(x-\bar{b})^\gamma} \mathrm{d}x$

I would like to find a "closed form" for the integral $$I(\beta) = \int_0^\infty \frac{x^\alpha(1-x)^\beta}{(x-c)^\gamma(x-b)^\gamma(x-\bar{b})^\gamma}\mathrm{d}x$$ where $\gamma=3/2$, $\alpha,\...
3
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178 views

Is this expansion related to Pochhammer's symbol, the gamma function, and beta function valid for $a<0$ and $a\neq 0,-1,-2,\ldots$?

If I start with the definition of the beta function $$B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} \operatorname{d}t = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}$$ that is valid for $\mathcal{R}(a) >0$ and $\...
3
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93 views

Log-Beta integral.

Is there a nice closed-form solution to the following integral: $$\int_0^1[\theta+\mu]^{a-1}[1-(\theta+\mu)]^{b-1}(e^{-\theta})^{S-1}(1-e^{-\theta})^{n-S}d\theta?$$ Here, $\mu,a,b>0$ and $S,n$ ...
3
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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\ \...
3
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0answers
160 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
3
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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 ...
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\...
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0answers
120 views

How is this formula for the Dirichlet $\beta$-function derived?

According to Wikipedia, we have: $${\displaystyle \beta (2k)={\frac {1}{2(2k-1)!}}\sum _{m=0}^{\infty }\left(\left(\sum _{l=0}^{k-1}{\binom {2k-1}{2l}}{\frac {(-1)^{l}A_{2k-2l-1}}{2l+2m+1}}\right)-{\...
2
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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 ...
2
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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
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0answers
144 views

When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there exist some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(...
2
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0answers
111 views

Parameterizing monotonic functions

I am trying to design a function $f_{\alpha} : \left[0,1\right] \rightarrow [0,+\inf)$ that looks like the following: It should have the following properties: Its integral should be 1 for any value ...
1
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0answers
29 views

Solving for probability from expressions involving incomplete beta function

For a given $\alpha,\beta,a,b>0$, I'm trying to find the value of $p$ that satisfies the following two equations $$\frac{1}{2}\bigg[a\bigg(b-\frac{\alpha}{\alpha+\beta+1}\bigg)+1\bigg] = I_p(\...
1
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0answers
31 views

Analogue to the beta-binomial distribution for sampling without replacement?

The beta-binomial distribution characterizes the number of successes in $n$ trials, but where the probability of success at each trial is unknown or random. However, suppose that you had finite ...
1
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0answers
36 views

Likelihood function with inequality

Suppose $Y_1, \dots, Y_n$ are i.i.d. bernoulli random variables. Also, $Y=\sum Y_i \sim {\rm binom}(n, \theta)$ and we have a prior beta distribution $\theta\sim {\rm beta}(a,b)$. I want to compute $P(...
1
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0answers
31 views

Correct interpretation + implementation of Beta Binomial sampling function?

I understand that the Beta Binomial distribution $BB(n,\alpha,\beta)$ can be interpreted as a Binomial distribution $Binom(n,p)$ where $p \approx Beta(\alpha,\beta)$. In some contexts we can interpret ...
1
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0answers
309 views

Consistency of MME estimator for Beta distribution

The Method of Moments' estimators for a beta distribution $B(a,b)$ are $$\hat{a}=m\frac{m(1-m)-v}{v},$$ and $$\hat{b}=\hat{a}\frac{1-m}{m},$$ where $m=\bar{X}$, and $v=\frac{n-1}{n}S_n^2$. Now, for ...
1
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0answers
61 views

Why the density function of Bernoulli trials where x is assumed from a beta distribution is the product of the binomial and the beta pdf?

I am trying to self-learning probabilities using the excellent Grinstead & Snell "Introduction to Probability" (that is also open source). I am stuck on Example 4.23, that involves a Bernoulli ...
1
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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 ...
1
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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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0answers
140 views

Is there any special property about incomplete beta function $B(x;1-x,x)$?

guys, I encountered a function like $$ f(x) = B(x;1-x,x) $$ where $B(\cdot)$ is the incomplete beta function and input $0 < x < 1$ is some positive small real value close to zero . I want to ...
1
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0answers
127 views

Riemann's zeta function in a definite integral

I am trying to evaluate the value of the definite integral $ I= \int_0^1((1-\delta)\log[p]-\delta\log[1-p])^4dp$ Using Binomial expansion I get $I=(1-\delta)^4\int_0^1(\log[p])^4dp-4(1-\delta)^3\...
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0answers
56 views

The integral $\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx$ and how simplify the Pochhmammer symbol in related series

Inspired in the shape of useful integrals to compute $\pi$ (see *), I've consider for each integer $k\geq 1$ $$\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx=\int_0^{\frac{1}{2}}x^{k-1}\sum_{n=0}^\...
1
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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\...
1
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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 ...
0
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0answers
10 views

Probability Beta distribution B1 less than Beta distribution B2

I am looking for the probability that a variable drawn from a Beta distribution $B_1$ with parameters $\alpha_1$, $\beta_1$ is less than a variable drawn from an independent other Beta distribution $...
0
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0answers
14 views

minimizing a double integral including beta function

I want to show that the argmin of the below integral is at $p,q =\eta$. ${1\over \beta(k(p+q),k(1-p-q)) \ \beta(kp,kq)} \iint\limits_{x+y \leq 1,\ x,y \geq \gamma} {x^{kp-1} y^{kq-1} (1-x-y)^{k(1-p-q)...
0
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0answers
26 views

Is a Beta distribution a continuous version of the Binomial Theorem?

The visual appearance of the PDF for a Beta distribution resembles that for the terms in the Binomial Theorem. Is the former a continuous variant of the discrete terms of the latter? Are they ...
0
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0answers
18 views

Beta-binomial distribution for scaled and translated Beta

Recall, that a binomial distribution in which the probability of success at each trial is randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can calculate ...
0
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0answers
44 views

Understanding The posterior distribution for a given model if it has some prior?

I was studying the posterior distribution and came across a question and didn't understand. What is the posterior distribution for a given that if a model has the following prior, $$𝑥_1, 𝑥_2,\dots,...
0
votes
0answers
12 views

The internally studentised residuals $r_i$

Suppose $Y = X \beta + \epsilon$ where $\epsilon \sim N(0, \sigma^2I)$. Show that for the internally studentised residuals $r_i$ defined as $$ r_i = \frac{\hat{\epsilon}_i}{\hat{\sigma}\sqrt{1-h_{...
0
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0answers
27 views

Upper bound for the complex Beta function

Is there any work or reference regarding upper bounds for the complex beta function defined by \begin{equation} B(x,y)=\frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}, \end{equation} for $\Re{x} >0$ and $...
0
votes
0answers
33 views

A conditional expectation of the beta binomial distribution?

Consider a beta binomial distribution where the number of trials, $n$, is odd and the shape parameters of the underlying beta distribution, $\alpha$ and $\beta$, are equal. Is there a closed form ...
0
votes
0answers
62 views

Proof of relation of Gamma and Beta distribution without using Jacobian

If X1 and X2 independent random variables follow Gamma Distribution, can we prove Y= X1/(X1+X2) is a Beta Distribution without using the Jacobian Change of Variable method? In our course, we haven't ...
0
votes
0answers
39 views

Sum of powers of regularized incomplete beta function

Let $k$ and $r$ be natural numbers, and $p$ is a rational number in $[0,1]$. Is it possible to compute exactly in closed form the following sum? $$ \sum_{i=1}^{\infty}I_{p}(i,k)^r, $$ where $I_x(a,b)$...
0
votes
0answers
14 views

Bounds of the second parameter of the incomplete beta function

Does the incomplete beta function $B(z,a,b)$ still hold even for negative values of $b$? For instance, consider $$\rho(r)=\frac{b_{0}}{1-q}B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$ ...
0
votes
0answers
40 views

Finding the inverse of an incomplete beta function

Is there a rigorous way of inverting $$\rho(r)=\frac{b_{0}}{1-q}B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$ where $B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$ is an ...
0
votes
0answers
91 views

Incomplete Beta function

I am looking for an approximation/bound for the incomplete Beta function $B_z(a,b)$ when $z\to0$. I know the Taylor expansion would help. However, I need a power series in $z^n$ (the exponential is an ...
0
votes
0answers
84 views

The expected value of Beta Function

Estimate the probability of success Suppose I send 10 tasks to my machine. 6 out of 10 tasks success, and 4 failed. These outcomes is summarized by $X$ as a binary variable, 1 is task success, and 0 ...
0
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0answers
17 views

Beta Distribution - Statistics

I am not sure if the process that I am currently using is statistically correct. I have some data that are arranged with a beta distribution. I need to generate randomly another set of data with the ...