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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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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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10 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_{...
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1answer
106 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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16 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
25 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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17 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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0answers
18 views

Find posterior distribution given beta prior

I have the following question: 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 ...
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0answers
22 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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1answer
74 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
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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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32 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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3answers
54 views

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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7 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 $$\rho(r)=\frac{b_{0}}{1-q}B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$ ...
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24 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
42 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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40 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 ...
5
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2answers
164 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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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
57 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
46 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}+\...
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4answers
197 views

Methods to attack integrals that include $(1+x)^{a}\ln^{b}(1+x)$ in the integrand

I am looking for systematic methods to attack the following class of integrals involving logarithmic functions $$\begin{aligned} I_{0} &= \int_{0}^{1}(1+x)^{a}\ln^{m}(1+x)\,\mathrm{d}x \\ I_{1} ...
11
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1answer
247 views

Solving $\int_0^{\frac{\pi}{2}}\frac{1}{\sin^{2n}(x) + \cos^{2n}(x)}\:dx$

Spurred on this question I decided to investigate the following integral: \begin{equation} I_n = \int_0^{\frac{\pi}{2}}\frac{1}{\sin^{2n}(x) + \cos^{2n}(x)}\:dx \end{equation} Where $n \in \mathbb{...
9
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1answer
275 views

Solving used Real Based Methods: $\int_0^x \frac{t^k}{\left(t^n + a\right)^m}\:dt$

In working on integrals for the past couple of months, I've come across different cases of the following integral: \begin{equation} I\left(x,a,k,n,m\right) = \int_0^x \frac{t^k}{\left(t^n + a\right)^...
2
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1answer
40 views

Fallacious moving of powers resulting with a correct trigonometric series identity.

Prove that $$ \\ \sum_{r=0}^n \left( \frac{ (-1)^r {n \choose r} } {n+r+1} ( \sin^{2(n+r+1)}x + \cos^{2(n+r+1)}x )\right) = \sum_{r=0}^n \frac{ (-1)^r {n \choose r} } {n+r+1} $$ for all values of ...
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3answers
52 views

Problem with evaluating $\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$ using Beta Function

Recently I've been trying to tackle the integral $\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$ using the Beta function $$\frac{B(\frac{x}{2},\frac{1}{2})}{2}=\int_0^{\frac{\pi}{2}} \sin^{x-1}(\...
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1answer
41 views

Beta distribution: find the parameter $\alpha$ of $\mathcal{B}e(\alpha,\frac{1}{3})$

I have this variable with beta distribution : $Y \sim \mathcal{B}e(\alpha,\frac{1}{3})$. I have to find the value of $\alpha$ such as : $P(Y \leq 0.416) =0.2 $ Formally for $\alpha \geq 0$ , $\beta \...
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1answer
63 views

How to evaluate this$ \int_{0}^{c} y^{\alpha-1}(1-y)^{\beta-1}dy$?

How do I evaluate the following integral? $$ \int_{0}^{c} y^{\alpha-1} (1-y)^{\beta-1}dy $$ where $1\geq c>0$. Thank you in advance.
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2answers
138 views

Seeking Methods to solve $F\left(\alpha\right) = \int_{0}^{1} x^\alpha \arcsin(x)\:dx$

I'm looking for different methods to solve the following integral. $$ F\left(\alpha\right) = \int_{0}^{1} x^\alpha \arcsin(x)\:dx$$ For $\alpha > 0$ Here the method I took was to employ ...
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0answers
26 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 ...
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0answers
16 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 ...
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3answers
66 views

Calculate limit of \Gamma function

Show that $$\lim _{x \to \infty} \log \left( \frac{ \sqrt{x} \Gamma\left(\frac{x}{2}\right) } {\Gamma \left( \frac{x+1}{2}\right)} \right) = \frac{1}{2} \log(2),$$ where $\Gamma$ is the Gamma ...
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1answer
43 views

Beta function $nB(\frac{3}{4}, n-1) \neq 1 $

I need to know if the equation $nB(\frac{3}{4}, n-1) \neq 1$ is true $\forall n \in\mathbb{N}$. I do not idea whether it is true or not. I was trying to prove it but I have not idea how to deal with ...
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1answer
63 views

Formula for the ratio $\frac{\Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n + 1)}$ of two values of the Gamma function

Show that $$\frac{\Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n + 1)} = \frac{1 \cdot 3 \cdot \cdots (2 n - 3) (2 n - 1)}{2 \cdot 4 \cdot \cdots (2 n - 2) \cdot 2n} .$$ I have proved that $$\Gamma\...
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0answers
112 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)-{\...
4
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1answer
76 views

Integral $\int_0^1 \frac{(1-x^4)^{3/4}}{(1+x^4)^2}$, may involve beta function

Evaluate the integral: $$\int_0^1 \frac{(1-x^4)^{3/4}}{(1+x^4)^2} dx$$ I have used substitutions like $x^4 = u$, or $x^{-4} = u$. After many hours, I came up with $\frac{2}{1+x^4} = u$ which reduces ...
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0answers
15 views

is there a specified way to get this expansion?

$\beta(x,y-1)=\beta(x,y)+\beta(x+1,y)+\beta(x+2,y)........$ i have tried to find the proof of this formula on google and some books but i found other methods like this and can not understand them can ...
2
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1answer
83 views

How do we prove that $\int_{0}^{\pi/2}\sin(t)^{2n+3}dt=\frac{4^n(2n+2)}{(2n+3)(2n+1){2n\choose n}}$?

I saw this integral in a paper on hypergeometric functions: $$S(n)=\int_{0}^{\pi/2}\sin(t)^{2n+3}dt=\frac{4^n(2n+2)}{(2n+3)(2n+1){2n\choose n}}\;\;\;\;\;\;\;\;\;\;\;(1)$$ I tried to prove it and got ...
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1answer
67 views

Finding a value from a probability density function

I've been asked the following based on a probability course I'm taking : $$f(x) = \begin{cases} cx^3(1-x)^2 & 0 \le x \le 1 \\ 0 & \text{otherwise.} \end{cases}$$ where $x$ has a beta ...
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2answers
65 views

Generalized Telescoping Series

Goog Morning Everyone, I'd like to ask the following exercise that my professor gave, that i think it has more theory behind it than expected it. The exercise ask to prove what the following ...
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100 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 ...
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1answer
145 views

Beta distribution CDF to Binomial Survival Function

There is a claim in my book that there is a connection to the Beta CDF and a Binomial Summation without explaining further. "Integration by Parts can be used to show that for $0<y<1$, and $\...
2
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2answers
106 views

How to evaluate this integral - beta function?

$$ \displaystyle\int\limits^{\cssId{upper-bound-mathjax}{1}}_{\cssId{lower-bound-mathjax}{0}} \left(1-\left(1-x^3\right)^\sqrt{2}\right)^\sqrt{3}x^2\,\cssId{int-var-mathjax}{\mathrm{d}x} $$ I have ...
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1answer
137 views

Median of beta distribution for alpha = beta

I've seen that the median of the Beta Distribution cannot be defined by a closed form analytic expression but in most sites I see people give that, when the parameteres alpha and beta are the same, ...
0
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2answers
128 views

prove that $\int^1 _0 \frac{x^{m-1}(1-x)^{n-1}dx}{(a+x)^{m+n}} = \frac{\beta(m,n)}{a^n(a+1)^m}$

$$\int^1 _0 \frac{x^{m-1}(1-x)^{n-1}dx}{(a+x)^{m+n}} = \frac{\beta(m,n)}{a^n(a+1)^m}$$ While solving the above integral I tried making $a + x = t$ as one of the substitution but it turned out to be ...
0
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0answers
12 views

Derivation of the beta-pass algorithm

I'm having trouble following along the derivation of this algorithm: I can't see how we get from (2.38) to (2.39). Does it have to do with conditional independence? I.e that $P(A,B|C) = P(A|C) * P(B|...
4
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2answers
396 views

Solve the integral using beta functions : $\int_0^1 \frac{(1-x^4)^{3/4}}{(1+x^4)^2}\,dx$

I was solving questions on beta and gamma functions and then I came across this question. $$ \int_0^1 \frac{(1-x^4)^{3/4}}{(1+x^4)^2}\,dx $$ Generally in questions of beta functions integrals of the ...
4
votes
2answers
355 views

Sum of series $\frac{ 4}{10}+\frac{4\cdot 7}{10\cdot 20}+\frac{4\cdot 7 \cdot 10}{10\cdot 20 \cdot 30}+\cdots \cdots$

Sum of the series $$\frac {4}{10}+\frac {4\cdot7}{10\cdot20}+ \frac {4\cdot7\cdot10}{10\cdot20\cdot30}+\cdots $$ Sum of series $\frac {4}{10}+\frac {4\cdot7}{10\cdot20}+ \frac {4\cdot7\cdot10}{...
4
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1answer
457 views

Sum of $1+\frac{1\cdot 3}{6}+\frac{1\cdot 3 \cdot 5}{6 \cdot 8}+\cdots \cdots$

Finding sum of $\displaystyle 1+\frac{1\cdot 3}{6}+\frac{1\cdot 3 \cdot 5}{6\cdot 8}+\frac{1\cdot 3 \cdot 5 \cdot 7}{6 \cdot 8 \cdot 10}+\cdots \cdots$ Try: We can write sum as $$ \mathcal{S} ...
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0answers
30 views

Sample medain of the vector l2 norm from uniform distribtion in n-dimensional unit hypersphere

I want to solve next task. Suppose we have random variable $X$ with uniform distribution in $n$ dimensional hypersphere of radius $1$. We have $N$ independent samples. I want to find median of the ...
1
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0answers
30 views

Likelihood function with inequality

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