Questions tagged [beta-distribution]

A two-parameter family of univariate distributions defined on the interval $[0,1]$.

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525
votes
15answers
216k views

What is the intuition behind beta distribution?

Disclaimer: I'm not a statistician but a software engineer. Most of my knowledge in statistics comes from self-education, thus I still have many gaps in understanding concepts that may seem trivial ...
55
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4answers
39k views

Regression for an outcome (ratio or fraction) between 0 and 1

I am thinking of building a model predicting a ratio $a/b$, where $a \le b$ and $a > 0$ and $b > 0$. So, the ratio would be between $0$ and $1$. I could use linear regression, although it doesn'...
38
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3answers
16k views

Distribution of scalar products of two random unit vectors in $D$ dimensions

If $\mathbf{x}$ and $\mathbf{y}$ are two independent random unit vectors in $\mathbb{R}^D$ (uniformly distributed on a unit sphere), what is the distribution of their scalar product (dot product) $\...
88
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7answers
149k views

Calculating the parameters of a Beta distribution using the mean and variance

How can I calculate the $\alpha$ and $\beta$ parameters for a Beta distribution if I know the mean and variance that I want the distribution to have? Examples of an R command to do this would be most ...
38
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7answers
34k views

Relationship between Binomial and Beta distributions

I'm more of a programmer than a statistician, so I hope this question isn't too naive. It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's ...
25
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6answers
16k views

Beta regression of proportion data including 1 and 0

I am trying to produce a model for which I have a response variable which is a proportion between 0 and 1, this includes quite a few 0s and 1s but also many values in between. I am thinking about ...
21
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1answer
11k views

Choosing between uninformative beta priors

I am looking for uninformative priors for beta distribution to work with a binomial process (Hit/Miss). At first I thought about using $\alpha=1, \beta=1$ that generate an uniform PDF, or Jeffrey ...
34
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7answers
60k views

Real-life examples of common distributions

I am a grad student developing an interest for statistics. I like the material over-all, but I sometimes have a hard time thinking about applications to real life. Specifically, my question is about ...
9
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2answers
3k views

Beta as distribution of proportions (or as continuous Binomial)

Beta distribution is related to binomial being also the distribution for order statistics. Probability mass function of binomial distribution is $$ f(k) = {n \choose k} p^k (1-p) ^{n-k} \tag{1} $$ ...
5
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1answer
4k views

Difference between standard beta and unstandard beta distributions?

What is the difference between standard beta and unstandard beta distributions? How to understand in an article if it is not described if it is standard or not?
6
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1answer
4k views

Determining beta distribution parameters $\alpha$ and $\beta$ from two arbitrary points (quantiles)

Suppose I have two points $(p_1,x_1)$ and $(p_2,x_2)$ where $p_i$ is a probability on the beta CDF and $x_i$ is a value on that same CDF. How would I go about determining the beta distribution shape ...
5
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1answer
3k views

Remove effect of a factor on continuous proportion data using regression in R

I have a data set of continuous proportions which depend on a fixed-effect factor, e.g.: ...
43
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5answers
20k views

Does the beta distribution have a conjugate prior?

I know that the beta distribution is conjugate to the binomial. But what is the conjugate prior of the beta? Thank you.
23
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5answers
10k views

Dealing with 0,1 values in a beta regression

I have some data in [0,1] which I would like to analyze with a beta regression. Of course something needs to be done to accommodate the 0,1 values. I dislike modifying data to fit a model. also I ...
13
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1answer
4k views

Distribution of the ratio of dependent chi-square random variables

Assume that $ X = X_1 + X_2+\cdots+ X_n $ where $X_i \sim N(0,\sigma^2)$ are independent. My question is, what distribution does $$ Z = \frac{X^2}{X_1^2 + X_2^2 + \cdots + X_n^2}$$ follow? I know ...
20
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2answers
6k views

Why exactly can't beta regression deal with 0s and 1s in the response variable?

Beta regression (i.e. GLM with beta distribution and usually the logit link function) is often recommended to deal with response aka dependent variable taking values between 0 and 1, such as fractions,...
14
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2answers
8k views

Beta distribution on flipping a coin

Kruschke's Bayesian book says, regarding the use of a beta distribution for flipping a coin, For example, if we have no prior knowledge other than the knowledge that the coin has a head side and ...
16
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2answers
33k views

Calculate the confidence interval for the mean of a beta distribution

Consider a beta distribution for a given set of ratings in [0,1]. After having calculated the mean: $$ \mu = \frac{\alpha}{\alpha+\beta} $$ Is there a way to provide a confidence interval around ...
9
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1answer
3k views

Do two quantiles of a beta distribution determine its parameters?

If I give two quantiles $(q_1,q_2)$ and their corresponding locations $(l_1,l_2)$ (each) in the open interval $(0,1)$, can I always find parameters of a beta distribution that has those quantiles at ...
4
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1answer
472 views

Can this statistic be shown not to be sufficient for $\theta$?

This problem comes from Casella and Berger, who do not rigorously demonstrate (in their solution key) that the statistic is not sufficient. Let $X_1,\dots,X_n$ be a random sample from a population ...
19
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3answers
10k views

What is the relationship between the Beta distribution and the logistic regression model?

My question is: What is the mathematical relationship between the Beta distribution and the coefficients of the logistic regression model? To illustrate: the logistic (sigmoid) function is given by $...
5
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2answers
2k views

Limit of n times Beta(1,n) variables when n goes to infinity

Wikipedia states the following: (https://en.wikipedia.org/wiki/Beta_distribution#Special_and_limiting_cases) $$\lim_{n \to \infty} nB(1,n) = \operatorname{Exponential}(1).$$ I'm having trouble ...
27
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1answer
5k views

How to construct a multivariate Beta distribution?

What is a multidimensional generalization of the Beta distribution, in compliance with the following specification? I am not looking for the Dirichlet distribution. I am looking for a generalization ...
7
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1answer
670 views

Is there a bivariate $\beta$ distribution I can fit to my data?

I am analyzing two dimensional data. After analyzing each dimension with the help of the fitdistrplus and logspline packages, they both fit the Beta distribution. Is it possible to analyze the two ...
3
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1answer
239 views

How does the maximum distance between adjacent values vary for increasing $n$

That is, when is the $\underset{n \to \infty}{\lim} \max (X_i-X_{i-1})\rightarrow 0$, where $1<i\leq n$, and $X_i\geq X_{i-1}$ and when is the limit $\neq 0$? The question supposes that the ...
5
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2answers
1k views

What is the distribution of the ratio between independent Beta and Gamma random variables?

What would be the distribution of the following equation: $$y = \frac{a}{(a+d)^2}$$ where $a, d$ $\sim$ $\Gamma(M,c)$. Additionally, $a$ and $d$ are independent random variables.
20
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1answer
25k views

How to interpret the coefficients from a beta regression?

I have some data that is bounded between 0 and 1. I have used the betareg package in R to fit a regression model with the bounded data as the dependent variable. My ...
16
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4answers
12k views

How to implement a mixed model using betareg function in R?

I have a dataset comprised of proportions that measure "activity level" of individual tadpoles, therefore making the values bound between 0 and 1. This data was collected by counting the number of ...
14
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3answers
2k views

Whence the beta distribution?

As I'm sure everyone here knows already, the PDF of the Beta distribution $X \sim B(a,b)$ is given by $f(x) = \frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1}$ I've been hunting all over the place for an ...
21
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3answers
974 views

Why is there -1 in beta distribution density function?

Beta distribution appears under two parametrizations (or here) $$ f(x) \propto x^{\alpha} (1-x)^{\beta} \tag{1} $$ or the one that seems to be used more commonly $$ f(x) \propto x^{\alpha-1} (1-x)^{...
12
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3answers
4k views

Understanding the Beta conjugate prior in Bayesian inference about a frequency

Following is an excerpt from Bolstad's Introduction to Bayesian Statistics. For all you experts out there, this might be trivial but I don't understand how the author concludes that we don't have to ...
10
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0answers
2k views

How to generate 2 correlated Beta random variables

I was wondering if it might be possible to generate 2 correlated $Beta$ random variables? In other words, I want to generate two Beta random variables which can be said to have come from two Beta ...
6
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2answers
5k views

Expected value of $1/x$ when $x$ follows a Beta distribution

Let $x$ have the probability density: $$f(x) = \frac{x^{\alpha - 1}(1-x)^{1 - \beta}}{\mathrm{B}(\alpha,\beta)}$$ where $\alpha,\beta$ are two positive parameters and $0 \le x \le 1$ is the domain ...
4
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1answer
11k views

Does my data come from a gamma or beta distribution? [closed]

I have data and I want to ascertain whether it is beta or gamma distribution. Once I know what the distribution is, how do I find out what the parameters are?
9
votes
3answers
691 views

Distribution of $XY$ if $X \sim$ Beta$(1,K-1)$ and $Y \sim$ chi-squared with $2K$ degrees

Suppose that $X$ has the beta distribution Beta$(1,K-1)$ and $Y$ follows a chi-squared with $2K$ degrees. In addition, we assume that $X$ and $Y$ are independent. What is the distribution of the ...
9
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1answer
6k views

Product of beta distributions

I am looking at trigger efficiencies, meaning that I have some device that fires on $k$ out of $n$ events. In the end I am interested in some estimate of the efficiency $\epsilon$ which is the ...
6
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1answer
2k views

What is the probability P(X > Y) given X ~ Be(a1, b1), and Y ~ Be(a2, b2), and X and Y are independent?

Given two independent variables with Beta distribution, $X \sim \text{Be}(a_1, b_1)$ and $Y \sim \text{Be}(a_2, b_2)$, how do you find the probability that the value of X is greater than the value of ...
9
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2answers
3k views

Is the beta distribution really better than the normal distribution for testing the difference of two proportions?

I'm working at an online agency, where we run a lot of AB testing in order to test differences in proportion between two groups (test vs. control). Standard practice in the industry for testing ...
5
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1answer
6k views

Pdf of $y = - \log(X)$ when $X$ is beta distributed The expected value of $Y$

I want find the PDF of $Y = - \log(X)$ and $X$ has a beta distribution. I found the below formula as the answer but i think there should be (1-ey)b-1 part should added to this. Is that correct ? I ...
5
votes
1answer
1k views

Conjugate beta / interpretation of the "continuous binomial" signal

Note: this question has significantly evolved, thanks to inspiring comments by Tim. Assume there is some "truth" $x\in[0,1]=Beta(1,1)$ that is signaled with some precision. I assume that the ...
3
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1answer
5k views

UMP of a Beta($\theta,1$) distribution

I need to find the UMP of a random sample of a BETA$(\theta,1)$ distribution. I know that the pdf of this problem is $$f(x;\theta)=\theta x^{\theta-1}=\theta e^{(\theta-1)\log{x}}$$ After some ...
14
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4answers
7k views

How can I (numerically) approximate values for a beta distribution with large alpha & beta

Is there a numerically stable way to calculate values of a beta distribution for large integer alpha, beta (e.g. alpha,beta > 1000000)? Actually, I only need a 99% confidence interval around the mode,...
8
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1answer
2k views

beta-regression accounting for residual spatial auto-correlation in R

I have data on the interval (0,1), that I model using beta regression with the betareg package in R. This works well. However, I ...
8
votes
1answer
2k views

Kernel density estimation on bounded support?

I was looking for some way to deal with boundary bias of kde in case of a unit interval. One example is an usage of Chen estimators (or Beta estimators; an example might be seen here: http://stats-www....
5
votes
2answers
5k views

sum and product rules of probability

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu $...
4
votes
2answers
388 views

Expectation of the ratio between Beta and Gamma random variables

Given \begin{equation}\label{eq:definition_of_z} \begin{split} \textbf{Z} = \left[\begin{array}{cccc} {z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\ {z}_{21} & {z}_{22} & \cdots & {...
4
votes
3answers
614 views

Proving transformations of two independent chi-squared random variables is equivalent to a Beta distribution

I came across the following in some old class notes of mine: if $\chi_{v_{1}}^{2}$ is independent of $\chi_{v_{2}}^{2}$ then $\frac{\chi_{v_{1}}^{2}}{\chi_{v_{1}}^{2}+\chi_{v_{2}}^{2}}\backsim ...
2
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2answers
10k views

Generating Beta distributions with Uniform generators

I can generate as many samples from one or more uniform distribution (0,1) as I wish. How can I use this to generate a beta distribution ?
0
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0answers
122 views

Beta-distribution: how to generate a peak at certain mean value with a control on variance in extrems

Following Distribution that has a range from 0 to 1 and with peak between them?, I generated a beta distribution that has a peak between 0 and 1 at the mean value. When the mean value is 0.5, I can ...
11
votes
2answers
737 views

Distribution of the exponential of an exponentially distributed random variable?

Let $X$ be an exponentially distributed random variable, that is, with density function $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ ($\lambda>0$), and cdf $F_X(x)=1 - e^{-\lambda x}$. What is the ...