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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.

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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 $...
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1answer
51 views

The value of the integral $\int _0^{\infty }\:2^{-\frac{x^2}{4}}dx$ [on hold]

Options are as follows: a)$\sqrt{\frac{\pi }{\ln \left(2\right)}}$ b) $\sqrt{\frac{1}{\ln \left(2\right)}}$ c)$\sqrt{\frac{3\pi }{\ln \left(2\right)}}$ d)$-\sqrt{\frac{1}{\ln \left(2\right)}}$
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1answer
52 views

SImplification of $I=\int_{x=0}^{\infty}x^{n-1}(\alpha-x)^{m}e^{-\mu x}dx.$

Let $n$ and $m$ be positive integers, $\mu$ be real positive and $\alpha$ positive real number. I would like to compute the following integral if there is close formula $$ I=\int_{x=0}^{\infty}x^{n-...
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0answers
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Inequality of regularized incomplete beta function

Which is the easiest way to prove that $b \to I_x(a,b)$ is an increasing function? Where, as usual, $I_x(a,b)$ is the regularized incomplete beta function.
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3answers
79 views

Prove $\frac{\partial}{\partial m}\text{B}(n,m)=-\text{B}(n,m)\sum_{k=0}^{n-1}\frac{1}{k+m}$

where $\ \displaystyle\text{B}(n,m)=\int_0^1 x^{n-1}(1-x)^{m-1}\ dx=\frac{\Gamma(n)\Gamma(m)}{\Gamma(n+m)}\ $is the beta function, defined over positive $\ n,m>0$. The point of this post is to ...
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1answer
65 views

Beta function and gamma function

I would like to ask if someone could help me with following equation. \begin{equation} \Gamma(m)\,\Gamma(n) = \int_{0}^{\infty}x^{m-1}e^{-x}\,dx\,\int_{0}^{\infty}y^{n-1}e^{-y}\,dy \end{equation} \...
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1answer
39 views

Asymptotic expansion of incomplete beta function

I would like to write down an asymptotic expansion in the $N\to\infty$ limit of the following incomplete beta function $$B\left(\frac{N}{N+1};N,p+1\right)=\int_0^{\frac{N}{N+1}}x^{N-1}(1-x)^p\,\text{...
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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)...
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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 ...
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2answers
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Peculiar (convergent?) definite integral

I have been trying to calculate the integral: $$\int_1^{\infty} \left(\frac{x^2}{\sqrt{x^4-1}}-1\right)dx$$ A hint is to multiply the whole integral by $x^{\lambda}$, calculate the two terms ...
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1answer
46 views

Beta function in Philip J. Davis׳ Essay

This question is about equation number (4) in Philip J. Davis’ Essay titled "LEONHARD EULER'S INTEGRAL: A HISTORICAL PROFILE OF THE GAMMA FUNCTION". In there it is stated by the author "Euler ...
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1answer
22 views

conditonal distribution question

For conditional distribution $$f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}$$ this is the basic definition I know about conditional distribution Consider n + m trials having a common probability of ...
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1answer
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Calculate $ \int_{0}^{\pi} \frac{dx}{\sqrt{3-\cos(x)}} $ [closed]

$$ \int_{0}^{\pi} \frac{dx}{\sqrt{3-\cos(x)}} $$ I need to calculate this using Beta \ Gamma functions. I have tried the substitution $2 +\cos(x) = t$
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1answer
42 views

Integral $\int_{a}^s\int_{a}^u\frac{1}{\sqrt{s-u}\sqrt{u-v}\sqrt{v-a}}dvdu$

Is it possible to get a closed form for $$\int_{a}^s\int_{a}^u\frac{1}{\sqrt{s-u}\sqrt{u-v}\sqrt{v-a}}dvdu\quad?$$ If we look at the simple integral it is related to Beta function, but for the ...
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1answer
32 views

What are the poles and zeros of the Euler Beta function?

For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ does the Euler Beta function $B(x,y)$ equal zero? For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ ...
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2answers
76 views

Solution of $\int_x^1y^{a-1}\left(1-y\right)^{b-1}dy = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$

When is $f=g$ on $(0,1)$ for $f = \int_x^1y^{a-1}\left(1-y\right)^{b-1}dy$ $g = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$ Let me show their graphs. They are small, so I multiplied ...
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1answer
91 views

Prove an transformation formula for Gauss hypergeometric function $_2F_1(a,b;c;z)$

In " Special functions: an introduction to classical functions of mathematical physics" by Nico M. Temme, at page 113 is reported this formula: $$_2F_1(a,b;c;z)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\...
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2answers
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How do I evaluate $\int_{0}^{t} x^{\alpha+k-1}(t-x)^{\beta+k-1} dx$?

Evaluate $\int_{0}^{t} x^{\alpha+k-1}(t-x)^{\beta+k-1} dx$ I find this very difficult to evaluate. Please help me. This is under the chapter of beta functions. But, I cannot see exactly where to use ...
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1answer
47 views

Mathematically prove that a Beta prior distribution is conjugate to a Geometric likelihood function

I have to prove with a simple example and a plot how prior beta distribution is conjugate to the geometric likelihood function. I know the basic definition as 'In Bayesian probability theory, a ...
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1answer
34 views

Equation involving difference of beta CDFs

Consider the expression $$I_{p}(\alpha,\beta+1) - I_{p}(\alpha+1,\beta) = c$$ where $I_p(a,b)$ is the regularized incomplete beta function. Question: Given $\alpha,\beta,$ and $c>0$, what is $p$? ...
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1answer
30 views

Writing beta function in terms of gamma functions (by substitution)

I'm going over my note and try to write the Beta function in terms of gamma functions. However, I just can't get (1.73) from (1.72). Even if I substitute t/(1-t) ...
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1answer
26 views

Solving for probability in equation involving Beta function

Consider the equation $$p^\alpha(1-p)^\beta = cB(\alpha+1,\beta+1)$$ where $c,\alpha,\beta>0$ and $B(\alpha,\beta)$ is the Beta function. Question: What is the probability $p$ as a function of $c,...
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1answer
100 views

Study this improper integral $ \int_0^1 \frac{dt}{\sqrt{t}\,\sqrt{1-t}\,\sqrt{1-\alpha\,\sqrt{1-t}}}$

I'm trying to study the behavior of this improper integral $$ \int_0^1 \frac{dt}{\sqrt{t}\,\sqrt{1-t}\,\sqrt{1-\alpha\,\sqrt{1-t}}}$$ for:$$\alpha>0 $$ While I just can't understand the ...
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1answer
37 views

evalute an integral with beta function 3

I need to evaluate the integral $$ I=\int_0^{\pi/4} \sin^4 \theta \cos^5 2\theta~d\theta$$ I am thinking of changing $2\theta$ to $u$ but then I have a problem with $\sin (u/2)$. Any help would be ...
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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(\...
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1answer
12 views

Beta function integral appearing in the $O(n)$-model

While studying the $O(n)$-model I found myself in the need of integrating $$\frac{\int_0^{\pi}\text{d}\theta\sin(\theta)^{n-2}\cos(\theta)^{2r}}{\int_0^\pi\text{d}\theta\sin(\theta)^{n-2}},$$ where $n\...
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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 ...
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1answer
92 views

Series of Beta Function

Solve: $$\int_0^\frac{\pi}{2}\frac{\sqrt{\sin{x}\cos{x}}}{\cos{x}+1}dx$$ I tried $$\int_0^\frac{\pi}{2}\frac{\sqrt{\sin{x}\cos{x}}}{\cos{x}+1}dx=\sum_{k=0}^\infty(-1)^k\int_0^\frac{\pi}{2}(\sin{x})^...
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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,...
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1answer
21 views

Understanding Beta distribution in prior belief distribution

I was looking at the Bernoulli Distribution and its relation to the Prior Belief Distribution. The equation is written as $$ \frac{x^{\alpha - 1}(1-x)^{\beta -1}}{B(\alpha, \beta)}. $$ I've also ...
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1answer
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Given an $X_1,…,X_n \sim Unif[0,1]$ i.i.d. sample, after ordering them, one of $X_k^*=x$, give a maximum likelihood estimation for $k$ using $x$.

The problem: The title sums up the problem pretty well, we have a sample from an independent idential (uniform) distribution (i.i.d.): $X_1, ..., X_n \thicksim Unif[0,1]$, and we only know one ...
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0answers
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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_{...
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1answer
140 views

Integral involving incomplete beta function

I have the following integral, $$\int_{0}^1x^{a-1}(1-x)^{b-1}B_x(c,d)dx$$ where $B_x(c,d) = \int_{0}^xt^{c-1}(1-t)^{d-1}dt$ is the incomplete beta function, and $a,b,c,d>0$. Question: Does this ...
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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 $...
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1answer
31 views

$\int_{-1}^2 (1+x)^{p-1}(1-x)^{q-1} dx$ = $2^{p+q-1}\beta{(p,q)}.$

Prove that: $\int_{-1}^2 (1+x)^{p-1}(1-x)^{q-1} dx$ = $2^{p+q-1}\beta{(p,q)}.$ I tried converting the integral into the standard forms of beta function: $\beta{(p,q)}=\int_{0}^1 (x)^{p-1}(1-x)^{q-...
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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 ...
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1answer
28 views

Find posterior distribution given beta prior

The output of a certain integrated-circuit production line is checked daily by inspecting a sample of $100$ units. Over a long period of time, the process has maintained a yield of $80$ percent, that ...
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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 ...
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2answers
104 views

Special Functions defined by Integrals.

There was this integral which caught my attention, when I was checking out The Applications of Beta and Gamma Functions. So, how can i prove the below, using change of variable? $$\int_{0}^{1}\frac{...
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2answers
50 views

Where can I find papers and research on the Alternating Hurwitz Zeta Function?

The function is as follows (I don't know it's name, it could be 'Generalised Dirichlet Eta Function') $$f(s,q)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(n+q)^{s}}$$
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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 $I_x(a,b)$...
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Evaluate $\sum\limits_{k=1}^{\infty}\frac{B(k,k)}{k}$ where $B$ denotes the beta function

How can I evaluate $\displaystyle\sum_{k=1}^{\infty}\frac{B(k,k)}{k}$ Here B is the beta function
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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 $$\rho(r)=\frac{b_{0}}{1-q}B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$ ...
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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 ...
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1answer
98 views

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family.

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family. I know that the beta distribution is $f(x; \alpha, \beta)={1\over B(\alpha, \beta)}x^{\...
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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 ...
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2answers
171 views

Seeking methods to solve: $\int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt$

Seeking Methods to solve the following two definite integrals: \begin{equation} I_(n) = \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt \qquad J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt \end{equation}...
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0answers
48 views

How to show: $\psi^{(0)}\left(\frac{1}{n}\right) - \psi^{(0)}\left(1 - \frac{1}{n}\right) = -\pi\cot\left(\frac{\pi}{n}\right)$ [duplicate]

Based on a result I found recently and in conjunction with methods I've observed on MSE I was able to show that: \begin{equation} \int_0^\infty \frac{ \ln(t)}{t^n + 1}\:dt = -\frac{\pi^2}{n^2} \...
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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 ...
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1answer
55 views

How to find integration of function, in form of hypergeometric function, given below?

I would like to prove the left side to right hand side which is in form of hypergeometeric function. Looking for your hints, suggestions and solultions. $$ \alpha_{1} \int_{0}^{1} (1-z)^{\alpha_{1}+\...