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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4
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1answer
71 views

Running through math confusion how to finish this integral using Beta function.

I am working on this easy integral but I want to use Beta Trigonometric Function to solve it. I know the answer is $\pi$ by using the half angle formula and help from u-substitution. $\int_{0}^{2\pi} \...
0
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1answer
44 views

Existence and calculation of a fractional integral

Let $b >0$ be arbitrary but fixed and let $\alpha >0$ if $$g_{\alpha}(x)=\frac{1}{\Gamma(\alpha)}\frac{1}{x^{1-\alpha}} \chi_{]0,b]}(x)$$ If $f:[0,b] \rightarrow \mathbb{R}$ is continous in [0,b]...
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0answers
64 views

How do I solve this integral with Gamma (probably) or beta functions?

$$\iint\cdots\int_{\mathbb R^n}\left(\prod_{i=1}^n x_i^m\frac{e^{-x_i^2}}{e^\frac{x_i}{2}\cosh\left(\frac{x_i}{2}\right)}\right)d^nx$$ I have managed to bring it to a form where the elements of the ...
0
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0answers
31 views

Solve $\sum_{n=0}^\infty \text P_{-n}^{-n}(z)$ and $\sum_{n=0}^\infty \mathsf P_{-n}^{-n}(z$) with Associated Legendre P functions of type $1$ and $3$

Here is a simple looking sum which should have an alternate form since it is just a double hypergeometric series with the Associated Legendre P function of the First (aka Second) Type $\text P_a^b(z)$ ...
2
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1answer
50 views

if $X$ ~ $Beta(a,b)$ then what is $P(X < \frac{1}{4})$?

I have tried many ways but I can not understand how to calculate that probability when a and b are not specified, I never had a problem in solving probability problems until now but this one I can not ...
3
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2answers
69 views

Proving Renyi's result on the order statistics of the exponential distribution

The Wikipedia article on order statistics mentions the following result on the order statistics of an exponential distribution with rate parameter, $\lambda$: $$X_{(i)} = \frac{1}{\lambda}\sum\limits_{...
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1answer
33 views

Proving the identity between Beta and Gamma functions using semi-group property of the Gamma.

Reading the answer here, it seems like the relation: $$B(t,s)=\frac{\Gamma(t)\Gamma(s)}{\Gamma(t+s)} \tag{1}$$ follows from the semi-group property of the Gamma distribution since "the integral ...
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0answers
37 views

Why don't we reparametrize the Beta so its parameters represent number of heads and tails?

The Beta distribution with parameters $a$ and $b$ can be thought as the posterior distribution of the probability of heads when we start with a flat prior and observe $a-1$ heads and $b-1$ tails. And ...
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0answers
46 views

Why can't I calculate the inverse of this function?

I have the following function: $$\frac{\mathrm{Beta}(a+1,x-a+1)^b}{\mathrm{Beta}(b+1,x-b+1)^a},$$ where $x>0$, $0.1<a<0.4$, $0.1<b<0.4$. When $a<b$, the function is concave down ...
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1answer
43 views

Question about beta function and definite integral expression.

This question is in a sense homework, as I am a math graduate from many years ago, going through some old text book and trying some questions. The question: Given the beta and gamma function ...
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0answers
40 views

Why is $\sum_{n\ge1}\frac{\text B_\frac n2(-n,n)}{n^a b^n }=-\sum_{n\ge1}\frac{\left(\frac 2n-1\right)^n}{n^{a+1}b^n}$ close to (reciprocal) integers?

Here is a possible closed form of a sum with tetration in it made by equating coefficients of the Incomplete Beta function. This question is inspired by: Closed form of $$\sum\limits_{n=1}^\infty \...
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0answers
37 views

How to solve this integral? (with beta function and cosine)

I need help to solve this integral, please. $$\int_0^{\pi/2}{\cos^x(\theta)\cos(y\theta)d\theta}$$ Thank you!
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1answer
61 views

Incomplete hypergeometric function

The integral representation of the Beta function $$ \mathrm{B}(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1-t)^{\beta - 1} \; \mathrm{d} t $$ is often generalised to give the incomplete Beta function, $...
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0answers
24 views

Closed form of $\sum\limits_{m=0}^\infty \frac{(a)_m\text B_p(b+m,c-b)z^m}{m!}$ with the Incomplete Beta function.

This question comes from related “sum of incomplete gamma function” problems. Also,there are many research papers trying to make the Hypergeometric Incomplete. Here is one confluent example from ...
6
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2answers
260 views

Beta regression

Consider some positive random variables $X^1, X^2$ and $Y\sim Beta(p; 1)$ where $p=\beta_1X^1 + \beta_2X^2$. We have a random sample $\{X^1_i, X^2_i, Y_i\}$. Now, estimating $\beta_1, \beta_2$ is not ...
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0answers
33 views

Question on Rudin's exercise 8.21

Things to remark: $\Gamma(x) = \int_0^{\infty} t^{x-1}e^{-t}dt.$ $\int_0^1 t^{x-1}(1-t)^{y-1}dt $ = $\frac {\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ = $B(x,y)$. I don't understand from where do $(102)$ ...
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1answer
50 views

Baby rudin 8.21

Things to remark before 8.21. $\Gamma(x) = \int_0^{\infty} t^{x-1}e^{-t}dt. $ If $x>0$ and $y > 0$ , then $\int_0^1 t^{x-1}(1-t)^{y-1}dt $ = $\frac {\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ = $B(x,y)$. ...
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0answers
54 views

Derive Stirling's Approximation for Beta Function

I am trying to derive a Stirling's approximation for the Multivariate Beta function. This does not seem very hard, but I cannot even derive the Stirling's approximation for the usual Beta function. ...
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0answers
20 views

Searching for an analytic solution for the Clopper und Pearson confidence interval of the binomial distribution

I want to calculate the symmetric 95% confidence interval of the binomial distribution using the Clopper and Pearson methode. From my understanding, this leads to solving the inverse beta function: $...
1
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1answer
70 views

Simplifying $\int_0^1 x^{a-1} (1-x)^{b-1} \, _2F_1\left(1,d;c+d+1;2-\frac{1}{x}\right) \, dx$

I have been working on a statistics project that hinges on the following function. $$ \int_0^1 x^{a-1} (1-x)^{b-1} \, _2F_1\left(1,d;c+d+1;2-\frac{1}{x}\right) \, dx$$ Is it all possible to remove the ...
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1answer
57 views

Integral similar to Beta function

I am trying to calculate the integral $$\int_{1}^{2} y^{k-1}(y-1)^{-1/2}dy,$$ where $k$ is positive integers. But the integral domain is a little different from the definition of the Beta function. I ...
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0answers
18 views

Relationship to the gamma function and generalized Incomplete Beta function

I have a question regarding the following property $$\text B_{(\alpha,\beta)}(a,b)= \int_\alpha^\beta t^{a-1}(1-t)^{b-1}dt=\text B_\beta(a,b)-\text B_\alpha(a,b)$$ this property may have something to ...
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0answers
17 views

How do I summarize several beta-distributions?

A beta-distribution is dependent upon two variables, alpha and beta. Let’s call them a and b, for the sake of my thumbs. Using these variables you can create a function which, when mapped in a ...
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3answers
413 views

How to evaluate the integral $I = \int_o^{\infty} \frac{x}{\sqrt{e^{2\pi\sqrt{x}}-1}}dx$?

$$I = \int_{0}^{\infty} \frac{x}{\sqrt{e^{2\pi\sqrt{x}}-1}}\,dx$$. I tried to solve by substituting $t = 2\pi\sqrt x \implies t^2 = 4{\pi}^2x \implies tdt = 2\pi^2dx$ $$I = \frac1{8\pi^4}\int_0^{\...
3
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2answers
128 views

I have a question about combination series. i don't understand that

$$ \sum_{r=0}^n \frac{(-1)^r}{n+r+1} {n \choose r} = \frac{ \sqrt{\pi}~ 2^{-2n-1} n ! }{ \left(n + \frac{1}{2} \right)! } $$ How to explain what is the left series become to right form? I calculated ...
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1answer
63 views

Surface area of n-dim hyperspherical cap idea

I want to calculate the surface area of an n-dim hyperspherical cap with radius r=1. I found S. Li's evidence, but I don't understand his idea: http://docsdrive.com/pdfs/ansinet/ajms/2011/66-70.pdf ...
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0answers
52 views

Closed form of beta distribution CDF when $\alpha + \beta = 1$

Is there a closed form of the regularized incomplete Beta function $I_x(\alpha, \beta)$ when $\alpha + \beta = 1$? A nice closed form based on arcsin exists for the case where $\alpha = \beta = 1/2$ (...
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0answers
52 views

Difference between finding asymptotics of Beta function through Laplace versus through Stirling

Im trying to find the asymptotic expression for the beta function $B(x,y) = \frac{ \Gamma(x) \Gamma(y)}{\Gamma(x+y) } $. Using Stirling's approximation $\Gamma(x) \sim \sqrt{\frac{2\pi}{x} } (\frac{x}{...
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0answers
58 views

Is there an integral representation of the multivariate beta function?

The Beta function is defined by- $\textstyle\displaystyle{B(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt}$ From this we can derive- $\textstyle\displaystyle{B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}}$ ...
3
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2answers
165 views

A series expansion for $\left(x^2\,B(x,x)\right)^\frac nx$ or $\frac{1}{(Γ(2x))^\frac nx}$ for $0\lt x\le 1, n\in\Bbb N$

Central Beta function Β(x). Imagine we wanted to find a series expansion for $\left(x^2\,B(x,x)\right)^\frac nx, n\in\Bbb N $ which would be particularly difficult because it uses the Beta function. ...
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0answers
28 views

Intuition of Beta distribution with less-than-one parameters

According to Wikipedia, Beta distribution models the distribution of parameter $p$ of Binomial distribution. Concretely, consider a Binomial distribution $$f(k;n,p)=\binom{n}{k} p^k (1 - p)^{n-k}$$ ...
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0answers
28 views

Show that for a fixed $q$ the Beta function $B(p,q)$ is log convex.

I want to show that for a fixed $q$ the Beta function $B(p,q)$ is log convex. I tried to show that the second derivative of the Beta function with respect to $p$ is larger than zero, i.e. that: \begin{...
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0answers
38 views

Good Lower bound on the Beta function

Let $m>0$, $r>0$ and $0< k < m$ be three positive real numbers. I want to find a tight lower bound of the special case of the Beta function, which is given by $$ B\left(\frac{m-k}{2}, \...
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1answer
74 views

Infinite series involving generalized binomial coefficients and double factorial

I recently discovered the following infinite series $\displaystyle \sum_{k=0}^{\infty} \binom{t}{k} (-1)^k \frac{(2k-3)!!}{(2k-2)!!} = \frac{2t \Gamma\left(t+\frac{1}{2}\right)}{\sqrt{\pi}{\Gamma(t+1)...
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2answers
94 views

Show that $\int_0^{\frac{\pi}{2}} \sqrt{\sin\theta}\mathrm d\theta \int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{\sin \theta}} \mathrm d \theta=\pi$

Show that $$\int_0^{\frac{\pi}{2}} \sqrt{\sin\theta}\mathrm d\theta \int_0^{\frac{\pi}{2}} \frac{1}{\sqrt{\sin \theta}} \mathrm d \theta=\pi$$ My book wrote that $$=\frac{\Gamma(\frac{3}{4})\Gamma(\...
6
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1answer
238 views

On a better solution for $\large {\int_{-x}^x\frac{dx}{\sqrt {1-x^2} e^{bx^2+ax}}}$

In this user’s question I was able to find out the following. It is for a bigger physics research problem having to do with comparing a Wigner function corresponding to a state of light for the user ...
2
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0answers
49 views

$\lim_{n\to\infty} \sum_{k=0}^{n}(-1)^k\frac{\binom{n}{k}}{ak+b}; a>0,b>0$

Calculate $$\lim_{n\to\infty} \sum_{k=0}^{n}(-1)^k\frac{\binom{n}{k}}{ak+b}, \qquad \tag{$a>0,b>0$}$$ My working: $$\begin{align}\lim_{n\to\infty} \sum_{k=0}^{n}(-1)^k\frac{\binom{n}{k}}{ak+b} &...
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0answers
67 views

Beta integrals substitutions

I am trying to evaluate the following two beta function integrals $$ \sum_{k=0}^\infty x^k B(k,5/2) = \int ^1_0 \left[\frac{(1-t)^{3/2}}{t}\right]\frac{1}{1-xt}dt $$ and $$ \sum_{k=1}^\infty x^k B(k+1,...
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1answer
38 views

The Beta function but with arbitrary integral bounds

In my research, I need to calculate $\int_a^bx^m(1-x)^ndx$ where $a<b$ are real numbers and $m,n$ are nonnegative integers. Before I derive it myself, is it written somewhere? I am looking for ...
0
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1answer
110 views

On reordering infinite product

In expressing $B(x,y)$ as an infinite product using the infinite product definition of $\Gamma(x)$, the book Special Functions by Andrews, et al doing this way : $$B(x,y)= \dfrac{\Gamma(x)\Gamma(y)}{\...
2
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1answer
93 views

How Lebesgue dominated convergence theorem works when upper bound of an integral is not a constant? (Beta vs Gamma functions)

I am reading through the book Special Functions written by Roy, et al. At page 5, the limit is justified by Lebesgue dominated convergence theorem (LDCT). I didn't know this theorem so spent about a ...
0
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1answer
37 views

beta distribution as ratio gamma distributions

I need a proof of this statement please: Let $Y_1$ and $Y_2$ be independent random variables, where $Y_1$ is gamma distributed with parameters $\alpha$ and 1 and $Y_2$ is gamma distributed with ...
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0answers
16 views

Conditional beta posterior for uniform priors

I have three different random variables $\theta_1, \theta_2, \theta_3$ . These random variables are actually parameters of binomial likelihood Assume that I have prior distribution of $\theta_2 \sim ...
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0answers
93 views

Arc length of $\cos(\ln(x)$) and $\sin(\ln(x))$ in closed form.

In this post of mine on the arc length of $cx^n$ I found a general formula for the arc length of the link name above. This formula cannot cancel the radicals and radicand as this may get rid of its ...
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1answer
23 views

Finding expectation using iterated expectation in a production line case

A factory produces bolts with a defective rate that changes randomly and independently from day to day but is constant throughout any given day. Let $p_i$ denote the defective rate on day i, and ...
1
vote
1answer
67 views

Integral of Joint Beta Distribution

I have a joint pdf of three independent beta RVs $\{\theta_1, \theta_2, \theta_3\}$ with separate parameters $\{(\alpha_1,\beta_1), (\alpha_2,\beta_2), (\alpha_3,\beta_3)\}$ I need to find pdf $g(\...
1
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2answers
120 views

How do I simplify $\Gamma\left(\frac{2021}{2}\right)$? [duplicate]

I was tasked to evaluate the Beta function $$B\left(\frac{2021}{2},\frac{1}{2}\right) \tag{1}$$ I've expanded this to $$\frac{\Gamma\left(\dfrac{2021}{2}\right)\Gamma\left(\dfrac{1}{2}\right)}{\Gamma(...
1
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1answer
50 views

Large $a$ asymptotics for the incomplete Beta function $B(z,a,0)$?

The incomplete Beta function is defined by $$ B(z,a,b) = \int_0^z dt\; t^{a-1} (1 - t)^{b-1} \ . $$ Suppose that I set $b=0$, and assume $a>0$ as well as $0 < z < 1$. My question is what are ...
2
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2answers
96 views

Find a lower bound for a Riemann integral.

Let $a,b$ be strictly positive constants and let $p\in(1/4,3/4)$. Is there $c>0$ such that $$\int_0^1(1-t^a)^{1/2+p}[2t-(2+b)t^{b+1}]dt>c \quad ?$$ *The constant $c>0$ may depend on $a,b$ or ...
0
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2answers
74 views

Calculate $H= \int_0^\infty \frac{x^{n+1}}{1+x^n}dx, \, n>2$

Calculate the following integral: $H=\displaystyle \int \limits_0^\infty \dfrac{x^{n+1}}{1+x^n}dx, \, n>2$ I wanna use Beta function $B(x,y)=\displaystyle \int \limits_0^\infty \frac{t^{x-1}}{(1+t)^...

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