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Questions tagged [beta-distribution]

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

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Distribution of product and sum of beta random variables

I have a set of probabilities that I am modelling as (independent) $p_{A,i} \sim Beta(\alpha_{A,i},\beta_{A,i})$ and $p_{B,i} \sim Beta(\alpha_{B,i},\beta_{B,i})$. I am calculating the following ...
Josh9999's user avatar
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Distribution of the correlation coefficient based on quadratic forms

Let $x,y$ be two independent random correlated vectors following the same multivariate (real or complex) centred normal distribution, and let $A$ be a non-negative linear operator. We can read here, ...
Alexandre's user avatar
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How to approach GLMs using data with beta distribution in R?

I'm trying to run some models on bee presence with five predictor variables. A snippet of the data is attached, but essentially I measured floral abundance and richness, calculated floral evenness and ...
alexia m's user avatar
5 votes
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Need help in calculating $\mathbb{E}(\frac{1}{x_{(2)}-x_{(1)}}\int_{x_{(1)}}^{x_{(2)}} f(t) \ dt)$, where $x_{(i)}$ are related Beta distribution

Suppose $Y, Z \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$. Let $a = g(\min(y,z)),\ b=g(\max(y,z)).$ How can I calculate the expectation $$\mathbb{E}\left[\frac{1}{b-a}\int_a^b f(t) \ dt\right]$$ ...
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constant approximation based on "sorted uniform distribution" and beta distribution [closed]

Let $X_1, X_2 \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$ and then sort $X_1,X_2$ to get $X_{(1)} < X_{(2)}$. Based on the pdfs of $X_{(i)}$, we know $X_{(1)} \sim \mathrm{Beta}(1,2)$ and $...
learner's user avatar
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2 answers
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Retrieving beta distribution parameters, alpha and beta, from mode and variance

Is it possible to decide the parameters of beta distribution ,alpha and beta, when mode and variance are given? I know the two parameters can be decided from mean and variance as: $E[X] = \dfrac{a}{a+...
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F test to compare variances. How is ncp calculated to obtain the F distribution under the alt hypothesis?

Recently I needed to do a power analysis for a comparison of variance with a F test. In the past I used the OC curves from Edwin Crow's "Statistics Manual" Chart VIII to get beta. https://...
TC1's user avatar
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How to resolve the residual versus predicted quantile devation (Dharma plot)?

I have been trying to perform beta regression modeling with random effects. I have sex ratio (0.561, 0.765 etc) as the response variable, and climatic variables + years (1970-2021) as predictor ...
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How to check if a set of numbers matches a Beta (4,4) distribution?

I have a random number generator that should generate values based on a Beta 4,4 distribution and some minimum and maximum. (IE, values between 0.9 and 1.1) I can generate a large set of numbers using ...
Tyler Shellberg's user avatar
5 votes
2 answers
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Why do many people like to use "normal distribution" instead of $\beta$ distribution when the theoretical range of data is finite?

Say we are approximating the distribution of the test score of a large group of students, while the theoretical minimum score is 0 and the maximum is 100. In this case, many people use the normal ...
Zuriel's user avatar
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Can Phi coefficient for intercept be negative?

I am working on beta regression model with two grouping variables (farm and years). Climatic variables are my predictors. Response variable is male proportion. I standardized all variables prior to ...
Rahul's user avatar
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How to calculate the expectation of the following Dirichlet distribution and Beta distribution?

This is a question from my research, related to the derivation of the variational EM algorithm with mean-field assumption about LDA-based model. We all know, given that $\boldsymbol{\theta} \sim \...
Henry Zha's user avatar
1 vote
1 answer
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Statistical test to compare beta distributions between two groups

I'm not sure if I'm using the right terminology but hopefully the code and plot will explain it well. I have six samples in two groups. Sample 1, 2 and 3 are in Group 1 and Sample 4,5,6 are in Group 2....
purple1437's user avatar
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How can one measure spatial clustering of continuous (non-count) data?

I am currently teaching an ecology class how to use the variance to mean ratio (VMR) as a method for looking at the spatial clustering of individual organisms across a landscape. This method takes the ...
Michael L's user avatar
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binomial GLM for proportion Vs Beta regression

Suppose that I have a dependent variable which is the proportion of persons infected with a certain disease out of the total number tested in different locations. Assuming the difference in ...
Wagathu's user avatar
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1 answer
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Random Sampling from Distribution

I have data of lengths ranging from 25 to 135 and I would like to determine the distribution of this data so that I can randomly generate values from this distribution. My data contains 443 values, ...
Hal's user avatar
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2 answers
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How good is the Beta distribution as a conjugate for Binomial distribution?

I understand that the Beta Distribution is a 'natural conjugate' of the Binomial distribution, in sense that the Posterior Distribution is proportional to the multiplication of both. $$ Posterior(\...
Oscar Flores's user avatar
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How to handle zero inflated data and incorporate random effect in beta regression?

My research question - how does the inbreeding level change in different age (life stages)? I have data of about 12000 observation done in different year (2007 to 2014) and 10 different islands (...
Nitya Shrestha's user avatar
4 votes
2 answers
303 views

Taking the limit of a Beta Distribution to yield the Gamma Distribution

The Poisson Distribution may be obtained from the Binomial Distribution by keeping $\lambda = np$ fixed and taking the limit as $n \rightarrow \infty$. Similarly, the Gamma Distribution may be ...
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Modelling a proportion/percentage which is left censored

I am looking at component wear type problem, where the dependent variable is a percentage of the original wall thickness. I had read on these forums that the use of a beta regression would make sense ...
Meep's user avatar
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Is this ratio between sum of squares beta distributed?

Consider $x_i \sim N(\mu,\sigma)$ I am interested in distribution of the the following statistic (arises from likelyhood ratio test): $$ \frac{ \sum_{n=1}^{n} (x_i - \overline{x})^2 }{ \sum_{n=1}^{n} (...
Bait Hoven's user avatar
4 votes
1 answer
163 views

Dependent variable is a bounded between 0 and 1

I need to test some hypotheses for a social sciences dissertation. In my description below, I refer to the independent as the Xs and the dependent variables as the Ys. I have jotted down what models/...
NutellaMonster's user avatar
3 votes
2 answers
104 views

Which distribution is this? [closed]

Context I'm working on a project where I need to understand the impact of some variables on satisfaction (y). My y variable is an NPS measure, ranging from 0 to 10 and does not have float values, only ...
João Bugelli's user avatar
10 votes
2 answers
617 views

Why is E(θ / (1 - θ)) different than E(θ) / (1 - E(θ))?

I've encountered a problem question: The probability of success for a random variable follows a Beta(5, 3) distribution. The posterior mean is θ = 0.625. The odds of success is defined as θ / (1 - θ)....
alexandrosangeli's user avatar
5 votes
1 answer
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Sufficient statistic for the family of PERT distributions?

A beta distribution is one of the form $$ \text{constant}\times x^{\alpha-1} (1-x)^{\beta-1} \, dx \quad \text{ for } 0<x<1. $$ According to this Wikipedia article, the family of "PERT ...
Michael Hardy's user avatar
1 vote
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What are the uniformly minimum variance unbiased estimators (UMVUE) for the minimum and maximum parameters of a PERT distribution?

I believe the answers to this question are the sample minimum and the sample maximum, but I have not been able to find a reference or proof of this.
Nick Stats's user avatar
2 votes
2 answers
320 views

What is the posterior probability for flipping a coin, assuming a beta distribution as conjugate prior

Suppose, I toss a fair coin n = 10 times and get 7 heads and 3 tails. The probability of fair coin is p = 0.5. Now, that the beta distribution is a conjugate prior of the binomial likelihood. I used ...
triangular_triffle's user avatar
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Can a softmax probability distribution be approximated to Beta distribution?

I am wondering if attention weights in BERT language model layers ( which ,in essence, are softmax probability distribution) can be approximated with beta distribution? With approximated beta ...
gray KK's user avatar
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6 votes
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Does this type of distribution have a name?

I have some integer data, produced by slightly convoluted numerical procedure, which is distributed between $0$ and $300$, with the most probable values being $0$ and $300$, and the least probable ...
Tor's user avatar
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1 answer
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Interpreting y axis in density plot

200 people were tested, 20 of those were infected. I want to get a posterior distribution for the uncertainty associated with the probability that a person is infected. I do this like this: ...
cvbzxc's user avatar
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Comparability of means for non-normal distribution and small sample size

I have to perform a comparability study between pre and post change of a production process. I'm using the final purity to measure comparability and wanted to compare medians between pre (n=40 runs) ...
sputch31's user avatar
3 votes
2 answers
92 views

Approximating the distribution of the product of iid beta variates

Background I am interested in the distribution of $$\theta_0=1-\prod_{i=1}^n(1-\theta_i)$$ where the $\theta_{i>0}$ are iid beta random variates: $$\theta_{i>0}\sim\text{Beta}(\alpha,\beta)$$ In ...
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2 votes
1 answer
336 views

How to interpret odds ratios by emmeans for glmmTMB-beta

I fit this mixed model with beta for the response variable: photochemical efficiency or Fv/Fm and the predictor variables are categorical: ...
Franelibethgalvez's user avatar
1 vote
1 answer
61 views

Model predictions are under-predicting on the high end of the distribution [duplicate]

I am trying to create a linear model where the dependent variable has the following summary features: Min 1st Qu. Median Mean 3rd Qu. Max 0.1579 0.3155 0.3547 0.3459 0.3827 0.4583 There were some ...
BSHuniversity's user avatar
5 votes
1 answer
158 views

Updating a Beta prior based on observations from a product of two Independent Bernoulli variables

I'm working on a problem involving Bayesian updating with a Beta prior, but the data I observe comes from a slightly complex source. Let $X \sim \text{Bernoulli}(p)$ and $Y \sim \text{Bernoulli}(q)$, ...
Parchment2382's user avatar
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1 answer
200 views

Fitting data with a beta distribution using fitdist from fitdistrplus package in R

I have a variable x that corresponds to data from a database. I've been trying to find the best distribution to fit it and looking at the histogram I figured either a normal, weibull or logistic ...
Ismael ancona's user avatar
3 votes
1 answer
60 views

Why does MCMC estimate the variance in a logit-normal model incorrectly?

I am trying to estimate the variance $\varepsilon^2_X$ in a simple logit-normal model of the form $ \sigma^{-1}(U) \sim \mathcal{N}(\mu_U, \varepsilon^2_U)$ $ \sigma^{-1}(X) \sim \mathcal{N}(\sigma^{-...
Eike P.'s user avatar
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1 vote
1 answer
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Bounded Distribution with specific limits regarding Variance

Im currently looking for a probabilty density function that posesses the following properties Should have range (0,1) $$ \lim_{\sigma \rightarrow 0} f(x) = \delta(1) $$ $$ \lim_{\sigma \rightarrow \...
elson1608's user avatar
2 votes
1 answer
39 views

Resulting beta distribution from two different samples

Let’s say each sample consists of 300 units inspected for defects. I have historic data from 100 samples in the past that give me an idea of what I expect the defect rate to be. I have a new batch to ...
cocoP's user avatar
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0 answers
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Mixed beta regression interpretation with categorical predictor

I have run mixed beta regressions on proportional data but I am struggling in my interpretation of the results. I understand this has been asked before but I have a categorical predictor with four ...
ScottM10's user avatar
3 votes
1 answer
267 views

Zero- and one-inflated beta GAMM (Generalized additive mixed model) in mgcv

I have vegetation cover (%) data [0,1] that includes 0's and 1's that I'd like to model with a beta GAMM, but don't understand the method for doing so. I've read that if the data includes 0's and 1's ...
Nate's user avatar
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1 vote
1 answer
488 views

How to interpret the DHARMa quantile residual plot?

We calculated a GLMM based on the beta distribution: ...
bos's user avatar
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0 votes
1 answer
105 views

Thompson Sampling with Two objectives - Cost and Success Rate

I have implemented a Thompson sampling algorithm with beta distribution that chooses between two processors to process the payments for each transaction such that it maximizes the success rate. For ...
Aayush Gupta's user avatar
0 votes
0 answers
118 views

Let X is beta distributed, what is the distribution of 1/X?

Let $X \sim Beta(a,b)$. I was wondering what is the distribution of $\frac{1}{X}$?. Here is my derivation by using transformation of random variable. Let $Y = h (X) = \frac{1}{X}$, which implies $h^{-...
Penn's user avatar
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0 answers
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Bounded variable: beta regression or switch to ratio?

My task is to study factors that influence the composition of labor force. The latter consists of two types of workers, full-time and part-time. My first approach was to run an OLS regression for the ...
Mikhail's user avatar
  • 97
2 votes
1 answer
122 views

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

Is there a similar way to this for calculating the $\alpha$ and $\beta$ parameters for a Beta distribution if I know the harmonic mean and variance that I want the distribution to have ($\alpha>1$ ...
peder's user avatar
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1 vote
0 answers
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Power of Uniform Order Statistics

I know that if $U$ is a uniform r.v. in $(0,1)$, then $U^a\sim Beta(1/a,1)$ with $a>0$. On the other hand, if $U_{(1)}\leq \cdots\leq U_{(n)}$ are the uniform order statistics, then, with $U_{(0)}=...
Pierre's user avatar
  • 111
2 votes
0 answers
65 views

Distribution of product of Beta and Chi-Squared?

If $X$ is distributed as Beta distribution and $Y$ as Chi-Squared, does the distribution of $Z = X Y$ have a name? For instance if $X\sim \text{Beta}\left(\frac{1}{2},1\right)$ and $Y\sim \chi^2(1)$, ...
Yaroslav Bulatov's user avatar
0 votes
0 answers
218 views

How to report emmeans results when p value = 1.00?

I am working with data of vegetation cover (proportions) for different height strata (0-5, 5-15, 15-30, >30 cm, and also bare ground) amongst four different sites and two different time periods (...
Agustín Gabriel Álvarez's user avatar
2 votes
1 answer
81 views

When can we assume a particular distribution in a statistical model?

In statistics, we often assume that a particular variable follows a certain distribution. For example, if we know $Y \in \{0, 1\}$, then we can assume $Y \sim \text{Bernoulli}(p)$, since using the ...
Sal Balkus's user avatar

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