Questions tagged [gamma-distribution]

A non-negative continuous probability distribution indexed by two strictly positive parameters.

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Can a mixed model random slope include a between-subjects factor in a repeated measures design?

I'm after some statistics help with generalised linear mixed models. I have built a model using glmer from the lmerTest package in R to fit my data with a gamma distribution. The formula is as follows:...
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confidence interval formula of gamma distribution? [closed]

i want a help can someone tell me the formula for confidence interval of gamma distribution? for alpha and beta Parameters i couldent find them? and there also this problem i dont know which pdf of ...
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Investigating the existence of plasticity (random intercept model vs. random slope model) in a response variable that is not normally distributed

I wanted to explore if the change in speed between day and night is similar among a set of individuals. To me, the logic way of testing this hypothesis is running a random intercept model and a random ...
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Cumulative Distribution function of ratio of 2 correlated gamma distributed random variables

Let $X \sim \Gamma(k_x, \theta_x)$, $Y\sim \Gamma(k_y, \theta_y)$ be correlated with correlation coefficient $\rho$. The shape and scale parameters are different. i.e. $k_x \neq k_y$ and $\theta_x \...
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Compounding Gamma with Gamma to yield F-distribution?

I am working through some problems from my Bayesian Statistics course and am having trouble understanding a step in the solution to a question. For reference this is the question: And here is the ...
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Issue about both results in agreement with 2 different ways to compute variance of a random variable : weighted chisquared vs Gamma distributions

1.) I am interested in computing the variance of this observable $O$ involving the coefficients of spherical harmonics $a_{\ell m}$ and the $C_{\ell}$ which is the variance of an $a_{\ell m}$ : $$O=\...
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Get probability distribution function from density function and calculate the cumulative value [duplicate]

For the given density function, how to find its distribution function and how to calculate the value of the distribution function? Density function: $$f(x) = \frac{1}{\Gamma(\frac{n}{2})}x^{\frac{n}{2}...
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How to build a regression equation for a gamma GzLM and how to interpret it?

I am trying to analyze if referral programs (1/0) have an impact on the average monthly spending of a user. I am confident that the gamma GzLM is the best model for my distribution: According to ...
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Renewal counting process with inter-arrival time gamma distribution: Model estimation

Let's start with the Poisson process: If $N_t$ is a Poisson process with parameter $\lambda$, then we know that the inter-arrival time distribution is an exponential distribution with parameter $\...
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Distribution for Fraction of Success in a Binomial Setting

So the actual original question I am trying to solve is a little bit different than the title: In a binomial setting with probability of success $p$, I keep examining observations until a fraction of ...
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Finding p-value given observation [closed]

Test the following hypothesis $$H_0: \lambda=\lambda_0$$ $$H_1: \lambda \neq \lambda_0$$ null distribution to be ~$gamma(20,20\lambda_0)$ given $\lambda_0=1$ and an observed sample =0.8, find the p-...
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Loss Size Index Function of A Gamma Random Variable

I'm trying to prove that the loss size index function of a Random Variable Y, which is distributed as a Gamma Random Variable ($Y \sim Γ(γ,c)$) has the following expression: $$ I(y) = \frac{\textit{G}(...
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The PDF of the Data (Marginal Likelihood) Given the Prior of a Gamma Distribution with Prior on the $ \beta $ Paraneter

Given a model where $ x_i | \beta \sim \mathcal{Gamma} ( \alpha, \beta ) $ where $ \beta \sim \mathcal{Gamma} ( \alpha0, \beta0 ) $, is there a closed form formula for the PDF of $ x_i $? Namely, what'...
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What order of power mean best estimates the median of a gamma distribution?

Suppose we have a gamma-distributed random variable $X$ whose shape/scale parameters are known to be $\alpha$ and $\beta$. What order $p$ for the sample power mean $\hat M_p[X]$ minimizes $$ (\mathcal{...
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Calculating the probability of the total duration of N sequential events with different cdfs describing their duration

Be patient, I am not very skilled with cdf. I seem to have a seemingly simple problem for which I either can't seem to find material about or simply lack the vocabulary for. Given are N sequential ...
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PDF of the exp-gamma distribution

exp-gamma distribution is defined as the density of the random variable log(X) when X is a gamma random variable. I am trying to obtain its PDF. Unfortunaltely, the only formula I have found is from a ...
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How to specify a parameters for Gamma distribution?

I have a task: A frequency claim distribution, $K$ is a compound Poisson-Gamma distribution. The mean of the Poisson distribution is gamma distributed with mean equal to 1 and variation equal 2. Find ...
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Generating function for Gamma distribution expectation

I encountered this formula in my assignment: $$X\sim \Gamma(\alpha, \beta), 1\le k < \alpha$$ $$ E(X^{-k})=\frac{\beta^k}{\prod^k_{i=1}(\alpha-i)} $$ And I wonder what would happen if $k$ is ...
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Choosing priors for the parameters of Gamma distribution

Suppose that $X_1, X_2, \cdots, X_n$ is a sample drawn from a Gamma distribution with parameter $\alpha$ and $\beta$. Then, the likelihood function can be written as follows: \begin{equation} L(\...
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Random numbers with exponentiated gamma distribution? [closed]

How to get random numbers following "exponentiated gamma distribution"? I tried to search some functions in R and this is what i got: https://rdrr.io/cran/Newdistns/man/expg.html I want to ...
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Intersection of multiple gamma distributions

Let's say I own a few hundred McDonalds locations. In a subset of those (say 100) I observe vegans eating there and I estimate the arrival time of vegans at these 100 restaurants using a Poisson ...
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Law of Total Variance

I trying to experiment with law of total variance in order to empirically recreate theoretical results. In particular I am interested in verifying that: $$ Var(Y) = E[Var(Y|X)] + Var(E[Y|X]) $$ Let's ...
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Help with interpretation of a zero-inflated gamma model

Background: I'm in fairly over my head here, working on a project for class where we had to choose our own publicly available dataset to analyse and I chose one with a zero-inflated continuous ...
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Can beta regression be used with continuous numerical data?

I am trying to fit a generalized linear model with Gamma distribution in R, but when I examine the residuals they are not normally distributed. I have a continuous numerical response variable, ...
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Help understanding gamma posterior of exponential likelihood

The posterior of $\text{Exp}(x;\lambda)$ with prior $\text{Gamma}(\lambda;\alpha, \beta)$ is $\text{Gamma}(\lambda|\alpha+n, \beta + n\bar x)$ where $n$ is the number of observations and $\bar x$ is ...
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Using GLM: Gaussian, Poisson vs Gamma

I am trying to perform a GLM analaysis using R for an outcome that is: Bounded by 0 - 10 In steps of 1 (Numerical Rating Scale for Pain: 0 - 10) I have a set of demographic factors, age, sex etc, ...
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Is there a closed form approximation for the composition of the Gamma CDF with the inverse Normal CDF?

Given $k$, $\theta$ fixed shape and scale parameters for some Gamma distribution which has a CDF $F$. Let $G^{-1}$ be the inverse CDF of the standard Normal distribution. Consider the composition $H(x)...
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How do we call the distribution of a ratio consisting of a constant divided by a truncated normal random variable?

I have been thinking for a while about the following problem of Cohen's d effect size measure $d={\frac {\mu _{1}-\mu _{2}}{\sigma}}$ in random-effects meta-analyses for which I cannot find a more ...
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Finding a convex upper-bound and lower bound on the expectation of $\ln(1+x)$ when $x\sim \text{Gamma}(k,\theta)$

I need to evaluate the expectation of $\ln(1+x)$ when $x\sim \text{Gamma}(k,\theta)$. I know that this integral $$\mathbb{E}\left[ \ln(1+x) \right] = \int_{0}^{\infty} \ln(1+x) \frac{x^{k-1}e^{-\frac{...
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Conflict between p-value and confidence interval from Gamma model

I am trying to fit the Gamma model with link = log in R using the glm function. Below is the summary of the model: The p-value for level 2 of modact_3 < 0.05, but the confidence interval for this ...
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Ratio of two independent Gamma distributions

Pre-requisites I have seen in Wikipedia that it says for independent r.v. $X \sim G(m), Y \sim G(n)$ the ratio $X/Y \sim B'(m,n)$. Where $G(x)$ denotes the standard Gamma distribution $$ f(x,\lambda)=...
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Predicting with a GLM

I wanted to check my understanding of predicting with a GLM: A binomial/logistic regression model predicts the binomial parameter = p = P(success). To convert the probability into classes, we have to ...
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How to use the gamma distribution to induce heteroskedasticity in the errors of a linear model

Suppose for $Y\in\mathbb{R}^n$, $X\in\mathbb{R}^{n\times p}$, $\beta\in\mathbb{R}^p$ and $\epsilon\in \mathbb{R}^n$, we have the linear model \begin{equation} Y=X\beta +\epsilon,\quad \epsilon\sim N(0,...
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Regression with Ratio of Frequencies as Response where some Frequencies Equal 0

I would like to perform a regression where my response is a ratio/proportion of two frequencies. Theoretically, the values of my response range from 0 to positive infinity. The frequencies represent ...
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Calculate mean for truncated Gamma distribution

I am using the "gamlss" and "gamlss.tr" packages in R to simulate truncated Generalized Gamma distributions The GG has three parameters mu, sigma and nu : see explanation https://...
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Negative Infinity AIC and BIC

I was trying to compare best fit model for monthly precipitation data sets and negative and positive infinity (-inf and inf) as values have showed up for both AIC and BIC tests. Can anyone tell me ...
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Performing post-hoc tests on a GLM with Gamma distribution

I am analyzing data that is gamma distributed. Hence, an ANVOA was a good choice but a GLM with gamma distribution worked well. To report the data I want to compare all groups to the control treatment ...
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Conditional gamma distribution derivation

Suppose we specify the gamma pdf in the following format: $f(x) = \lambda e^{-\lambda x} \frac{(\lambda x)^{n - 1}}{(n - 1)!}$ Further suppose we want the distribution of a $\text{gamma}(\lambda = 1,n ...
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Properties of $\chi^2(1)$ multiplied by a real value "$a$"

Is it true that the $\chi^2$ distribution with $k=1$ (noted $\chi^2(1)$) multiplied by a real value $a$ is equal to $\chi^2(a)$ ? If not, is there a particular distribution for $a\cdot\chi^2(1)$? If ...
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Overdispersion strategies for exponential curve fitting GLM?

I have price-volume (demand curve) data that follows the exponential function. I used a Bayesian model via PyMC3 to visualize the posterior distribution. This model is simply linear regression in log ...
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Relationship between Poisson and Gamma Distribution

There are several solutions to this problem but I am interested in the solution in Casella & Burger Pg. 100. The problem shows that if $X$ follows gamma($\alpha$, $\beta$), a random variable and $...
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Consistency when we want to find the distribution of sum of random variables following each one a distribution

I want to clarify a point that disturbs me among different cases. I am interested in formulate correctly in a general case when we know the distribution of different random variables and we want to ...
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Function fit to skewed data and non-zero beginning of the function

I would like to find a function that would represent the best fit to represent this type of biological data. More precisely, I would like to estimate expected daily egg production by an insect, based ...
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95 percentile for failure frequency with Poisson process

I am trying to recreate the stats for 'failure frequency' - in this case the 95 percentile - shown in the attached images below. For one set of failure data there are 27 failures over a cumulative ...
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How do I sample Simultaneous Sums of Gamma-Distributed Variables?

Suppose I have 7 variables $y_i$ sampled from $Gamm(a,1)$, with $a>0$. Now, I define $$x_1 = y_1+y_2+y_3+y_4,$$ $$x_2 = y_1+y_2+y_5+y_6,$$ $$x_3 = y_1+y_3+y_5+y_7$$ What is the distribution of $x_1$...
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How to calculate standard error given mean and confidence interval for a gamma distribution?

I am a health economist. I often look at information on costs, which is generally assumed to come from a Gamma distribution because costs are constrained to be zero or positive. Typically this ...
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Expected value of 1/(1−X) of a Gamma distribution [duplicate]

I was interested in calculating $E\left(\dfrac{1}{1-X}\right)$ where $X\sim$ Gamma ($n,\lambda$), but I wasn't able to solve the associated integral using standard integration techniques. $$E\left(\...
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Parametric test if underlying assumption is gamma distribution

I'm trying to concudt an A/B test where I want to compare the mean of two groups of customer spendings and decide whether they are significant or not. I have a dataset of 10 000 customers and their ...
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Finding a distribution of Accelerated Failure Model

Can someone please answer these 3 questions related to AFT model? In Accelerated Failure Time (AFT) model, $S(t│β,x)=S_0 (exp(β^T x).t)$, where $S_0$ is the baseline survival function, does $S_0$ ...
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Posterior distribution of $\theta x^{\theta - 1}$ with $Gamma(\alpha, \lambda)$ prior

Random variables $X_1, \ldots, X_n$ are i.i.d given $\vartheta = \theta$ and have the following pdf: \begin{equation} p(x|\theta)=\begin{cases} \theta x^{\theta - 1}, & \text{if $0<x<1$...
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