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 ...
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 ...
56
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4answers
40k 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'...
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.
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 ...
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) $\...
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 ...
29
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1answer
846 views

Jaynes' $A_p$ distribution

In Jaynes' book "Probability Theory: The Logic of Science", Jaynes has a chapter (Ch 18) entitled "The $A_p$ distribution and rule of succession" in which he introduces the idea of ...
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 ...
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 ...
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 ...
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 ...
21
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3answers
976 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)^{...
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 ...
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,...
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 $...
18
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1answer
23k views

Beta distribution fitting in Scipy

According to Wikipedia the beta probability distribution has two shape parameters: $\alpha$ and $\beta$. When I call scipy.stats.beta.fit(x) in Python, where ...
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 ...
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 ...
16
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3answers
2k views

What distribution does the inverse normal CDF of a beta random variable follow?

Suppose you define: $$X\sim\mbox{Beta}(\alpha,\beta)$$ $$Y\sim \Phi^{-1}(X)$$ where $\Phi^{-1}$ is the inverse of the CDF of the standard normal distribution. My question is: Is there a simple ...
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 ...
14
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1answer
6k views

Is the Gaussian distribution a specific case of the Beta Distribution?

If you look at a beta distribution with $\alpha=\beta=4$ it looks very similar to a Gaussian distribution. But is it? How can you prove whether a Beta(4,4) distribution is Gaussian or not?
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 ...
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,...
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 ...
12
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2answers
1k views

Why is this distribution uniform?

We are investigating Bayesian statistical testing, and come across an odd (to me atleast) phenomenon. Consider the following case: we are interested in measuring which population, A or B, has a ...
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 ...
12
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2answers
3k views

Efficiently sampling a thresholded Beta distribution

How should I efficiently sample from the following distribution? $$ x \sim B(\alpha, \beta),\space x > k $$ If $k$ is not too big then rejection sampling may be the best approach, but I am not ...
12
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1answer
4k views

Dealing with regression of unusually bounded response variable

I am attempting to model a response variable that is theoretically bounded between -225 and +225. The variable is the total score that subjects got when playing a game. Although theoretically it is ...
11
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1answer
2k views

Since the beta distribution is similar in form to the binomial, why do we need the beta distribution?

It appears that the binomial distribution is very similar in form to the beta distribution and that I can re-parametrize constants on either pdf to make them look the same. So, why do we need the beta ...
11
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2answers
741 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 ...
11
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1answer
1k views

Outlier detection in beta distributions

Say I have a large sample of values in $[0,1]$. I would like to estimate the underlying $\text{Beta}(\alpha, \beta)$ distribution. The majority of the samples come from this assumed $\text{Beta}(\...
11
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0answers
1k views

Hyperprior Noninformative Beta Binomial Model [closed]

I've been working through Gelman's Bayesian Data Analysis 3 text and have been trying to understand one of the hierarchical models revolving around rat tumors (Chapter 5). He uses a binomial model ...
11
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2answers
2k views

Scaling the backward variable in HMM Baum-Welch

I am just trying to implement the scaled Baum-Welch algorithm and I have run into a problem where my backward variables, after scaling, are over the value of 1. Is this normal? After all, ...
10
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2answers
592 views

Square root of a Beta(1,1) random variable

If $X^2 \sim \text{Beta}(1,1)$, is there a closed form for the distribution of $X$? If yes, what does it look like? And if this is not too much to ask, is there a general way to find the distribution ...
10
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1answer
3k views

Jeffreys' prior for Beta distribution

If my likelihood has the form of a beta distribution, and I want to use Jeffreys' prior for its parameters, what is form of the prior? For some distributions its pretty straight forward to calculate. ...
10
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2answers
3k views

UMVUE of $\frac{\theta}{1+\theta}$ while sampling from $\text{Beta}(\theta,1)$ population

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from the density $$f_{\theta}(x)=\theta x^{\theta-1}\mathbf1_{0<x<1}\quad,\,\theta>0$$ I am trying to find the UMVUE of $\frac{\theta}{1+\theta}$...
10
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1answer
369 views

If $X_1,X_2$ are independent beta then show $\sqrt{X_1X_2}$ is also beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities $\beta(n_1,n_2)$ ...
10
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2answers
2k views

Sampling distribution of the mean of a Beta

Say we have $X \sim \text{Beta}(\alpha, \beta)$. What's the sampling distribution of its sample mean? In other words, what distribution does the sample mean $\bar{X}$ of a Beta follow?
10
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1answer
2k views

Order statistic for beta distribution

Let $x_1,\dots,x_n$ be i.i.d. draws from $Beta\left(\frac{k}2,\frac{k-p-1}{2}\right)$. How are the minimum and maximum order statistics distributed, respectively? I would greatly appreciate a ...
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 ...
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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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 $$ f(k) = {n \choose k} p^k (1-p) ^{n-k} \tag{1} $$ ...
9
votes
1answer
265 views

Closed form for $\mathbb{E}[\ln (1-p)]$, for $p \sim Beta(\alpha, \beta)$

We know that if $p \sim Beta(\alpha, \beta)$, then $$ \mathbb{E}[\ln p] = \psi(\alpha) - \psi(\alpha + \beta) $$ where $\psi(.)$ is the Digamma function. Is there an easy form for $ \mathbb{E}[\ln (...
9
votes
1answer
602 views

Why is the Beta Distribution Called the Beta Distribution?

In my research, I've been unable to find a historical explanation for why the Beta distribution has the name it does. I'm aware of what Wikipedia says about how it got its name, but so far I haven't ...
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 ...
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 ...
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 ...
9
votes
2answers
434 views

Why does the proportion of native language speakers have an arcsine like distribution?

Based on actual data, given below is the distribution of the languages spoken in India by nearly $1.4$ billion people. There are more than $1600$ active languages in India which have been classified ...
9
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1answer
1k views

Logarithm of incomplete Beta function for large $\alpha,\beta$

R's function pbeta is supposed to calculate the incomplete regularized Beta function. If the flag log=TRUE is passed as an ...

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