Questions tagged [beta-distribution]
A two-parameter family of univariate distributions defined on the interval $[0,1]$.
116
questions
525
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15answers
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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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
1answer
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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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
$$ f(x) \propto x^{\alpha-1} (1-x)^{...
12
votes
3answers
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Understanding the Beta conjugate prior in Bayesian inference about a frequency
Following is an excerpt from Bolstad's Introduction to Bayesian Statistics.
For all you experts out there, this might be trivial but I don't understand how the author concludes that we don't have to ...
10
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0answers
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How to generate 2 correlated Beta random variables
I was wondering if it might be possible to generate 2 correlated $Beta$ random variables?
In other words, I want to generate two Beta random variables which can be said to have come from two Beta ...
6
votes
2answers
5k views
Expected value of $1/x$ when $x$ follows a Beta distribution
Let $x$ have the probability density:
$$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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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}}$$
After some ...
14
votes
4answers
7k views
How can I (numerically) approximate values for a beta distribution with large alpha & beta
Is there a numerically stable way to calculate values of a beta distribution for large integer alpha, beta (e.g. alpha,beta > 1000000)?
Actually, I only need a 99% confidence interval around the mode,...
8
votes
1answer
2k views
beta-regression accounting for residual spatial auto-correlation in R
I have data on the interval (0,1), that I model using beta regression with the betareg package in R. This works well. However, I ...
8
votes
1answer
2k views
Kernel density estimation on bounded support?
I was looking for some way to deal with boundary bias of kde in case of a unit interval. One example is an usage of Chen estimators (or Beta estimators; an example might be seen here: http://stats-www....
5
votes
2answers
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sum and product rules of probability
I am reading Bishop's Pattern Recognition and Machine Learning.
In page 73, chapter 2.1. I can't understand the formula 2.19 :
$$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
votes
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
votes
0answers
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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 ...
