Questions tagged [gamma-distribution]

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

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Distribution function

Find (without using MGF) the mean and variance. $$f(x) = \exp(-kx)x^{(r-1)}k^r/(r-1)!\ \text{ for }\ x>=0$$ $$f(x) = 0\ \text{ for }\ x<0$$ $r$ positive integer, $k>0$
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MLE for Gamma Shifted Distribution

I need to fit a gamma distribution that is shifted to the left and truncated at zero (so that for example, my data may only come from the right tail of the full distribution, and I don't have any ...
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How to validate & diagnose a gamma GLM in R?

I am fitting a generalized linear model in R with the log link and I need to validate and diagnose my model. I have never worked with the GLM in the past. Is there an article or any references I can ...
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Which diagnostics can validate the use of a particular family of GLM?

This seems so elementary, but I always get stuck at this point… Most of the data I deal with are non-normal, and most of the analyses based on a GLM structure. For my current analysis, I have a ...
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How can I estimate parameters of a convolution of an exponential and gamma?

Ideally, I would input a one-dimensional array of data, and output the estimates for the three parameters. I'm not very familiar with any statistics software but I have MATLAB (and all of the free ...
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Farlie-Gumbel-Morgenstern Bivariate Gamma Distirbution

Given the variables $X$ and $Y$, which are correlated, $X\ge0$, $Y\ge0$ and each follow a gamma distribution with different shape parameters, i.e.,$X\sim Gamma(a_1,\alpha)$ and $Y\sim Gamma(a_2,\alpha)...
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Choosing between LM and GLM for a log-transformed response variable

I'm trying to understand the philosophy behind using a Generalized Linear Model (GLM) vs a Linear Model (LM). I've created an example data set below where: $$\log(y) = x + \varepsilon $$ The ...
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Is there a distribution appropriate for a continuous variable skewed toward zero and able to include zero?

I am interested in modelling the impact of some environmental parameters on a concentration of measured phytoplankton pigment. The concentration of pigment is skewed so that low concentrations are ...
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How to use the SD of a normal sampling distribution to specify the gamma prior for the corresponding precision?

The gamma distribution is a commonly used prior distribution for the precision ($1/sd^2$) of a normal distribution in Bayesian hierarchical modeling. I want to use an informed prior for the variance ...
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Density of Y = log(X) for Gamma-distributed X

This question is closely related to this post Suppose I have a random variable $X \sim \text{Gamma}(k, \theta)$, and I define $Y = \log(X)$. I would like to find the probability density function of $...
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Specifying the Form of Prior, Likelihood and Posterior Distributions for Bayesian Analysis

I have recently begun to look into Bayesian Analysis, and, although I'm beginning to get to grips with the general framework (i.e. $\text{posterior} \propto \text{likelihood} \times \text{prior}$), I'...
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How do I establish shape and scale parameters?

I am trying to forecast customer LTV using an exponential gamma distribution suggested in a Journal of Forecasting article (Empirical Comparison of New Product Trial Forecasting Models; authored by ...
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Sum of Gamma distributions

The lifetime of a particular brand of batteries is known to have a gamma distribution. Tests on a large sample of these batteries show a mean lifetime of 480 hours and a standard deviation of 96 hours....
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Poisson is to exponential as Gamma-Poisson is to what?

A Poisson distribution can measure events per unit time, and the parameter is $\lambda$. The exponential distribution measures the time until next event, with the parameter $\frac{1}{\lambda}$. One ...
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The relationship between the gamma distribution and the normal distribution

I recently found it necessary to derive a pdf for the square of a normal random variable with mean 0. For whatever reason, I chose not to normalise the variance beforehand. If I did this correctly ...
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Construction of Dirichlet distribution with Gamma distribution

Let $X_1,\dots,X_{k+1}$ be mutually independent random variables, each having a gamma distribution with parameters $\alpha_i,i=1,2,\dots,k+1$ show that $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$,...
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Is left-of-mean cdf of exponential distribution greater than that of a gamma distribution?

Edited to restrict one of the gamma distributions to be an exponential distribution, and to force both means to 1.0, and to add wolfram alpha links: I have two distributions, both with mean 1, and I ...
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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 ...
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If $X \sim \mathrm{Gamma}(\alpha,1)$ and $Y \sim \mathrm{Gamma}(\beta,1)$, is $X/(X+Y) \sim \mathrm{Beta}(\alpha,\beta)$?

According to Wikipedia : $\mathrm{Beta}(\alpha,\beta) = \mathrm{Gamma}(\alpha,\theta) / (\mathrm{Gamma}(\alpha,\theta) + \mathrm{Gamma}(\beta,\theta))$ However, when I try to simulate this in R :...
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Gamma distribution and Cramér-Rao bound

There are two definitions of the GAMMA distribution: I did the ML estimation, generated the Fisher Information, compared it to the Variance and the Cramer Lower Bound was reached, so the estimator ...
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Maximum Likelihood Estimation of Inverse Gamma Distribution in R or RPy

I am trying to fit a three parameter inverse gamma distribution to my data in either R or Python. I would like to do this using maximum likelihood estimation (MLE). The pdf of the three parameter ...
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Posterior derivation with prior as Normal-Gamma distribution [duplicate]

I am studying Gary Koop's Bayesian Econometrics , it is bit confusing in the beginning. I will reproduce some results from the textbook here in order to smoothly move to my question. For a simple ...
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A question about parameters of Gamma distribution in Bayesian econometrics

The Wikipedia article on the Gamma distribution, lists two different parameterisation methods, one of them frequently used in Bayesian econometrics with $\alpha>0$ and $\beta>0$, $\alpha$ is ...
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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?
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Sum of exponential random variables follows Gamma, confused by the parameters

I've learned sum of exponential random variables follows Gamma distribution. But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what ...
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How to obtain the gamma distribution through convolution of two different distributions?

How can one obtain the gamma distribution through convolution of two different distributions? Could the gamma distribution be created as a non-trivial sum of $N$ random variables $X$ which have the ...
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How to calculate the boundary value for a random variable which is sum of variables with gamma and uniform distributions?

The variable is a sum of two random variable which obey gamma and uniform distributions, respectively. The parameters of the uniform distribution variable are determined, and the other's must be ...
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How does one get the gamma distribution parameters from a word problem description?

I am specifically confused as to the meanings of the shaping and scaling parameters of the gamma distribution and how they are used in a real context. Once I find the correct parameters, I have the ...
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McKay's bivariate Gamma distribution

Given the variables $X$ and $Y$, which are correlated, $X\ge0$, $Y\ge0$ and each follow a gamma distribution with different shape parameters, i.e.,$X\sim\Gamma(a_1,\alpha)$ and $Y\sim\Gamma(a_2,\alpha)...
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Confusion over Gamma distribution CDF

Here's a probability question (probably really simple) I'm not sure how to solve: Gamma distribution $X\sim \mathcal{G}(\alpha,\beta)$ with $\mu = 20$ and $\sigma^2 = 80$ $P(X \le 24)$ = ? The ...
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Joint distribution of two gamma random variables

I am so puzzled by this problem. Given two variables $X_1$ and $X_2$, such that $X_i \sim \mathrm{Gam}(a_i, b)$, find the joint distribution of $X_1$ and $X_2$. I understand how to proceed if ...
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How to test whether a sample of data fits the family of Gamma distribution?

I have a sample of data which was generated from a continuous random variable X. And from the histogram I draw using R, I guess that maybe the distribution of X obeys a certain Gamma distribution. But ...
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Bivariate Gamma distribution PDF

I'm analyzing a set of data, and I like to fit a gamma distribution. I know how to do it in one dimension, but the data that I'm analyzing now are two dimensional. Is there any way that I can have a ...
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Transformation to fit gamma distribution for glm

The data simulated below has a maximum value of 4 and is interestingly skewed. The maximum of 4 is a limitation imposed by the instrument used and the data is semi-discrete, i.e., there are a ...
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Proper use and interpretation of zero-inflated gamma models

Background: I am a biostatistician presently wrestling with a dataset of cellular expression rates. The study exposed a host of cells, collected in groups from various donors, to certain peptides. ...
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What reference can I cite for the proof that the sum of n exponential variables follows a gamma distribution?

There is a fairly common theorem, which states that: The sum of $n$ independent variables following an exponential distribution $\mathrm{Exp}(\alpha)$ follow an gamma distribution $\mathrm{Gamma} (n, ...
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A mixed model with gamma distribution: handling zeros

I have a panel cost data with some zero values (not missing values but rounded to 0). How should I handle zeros when I use SAS Proc Glimmix with Gamma distribution? Maybe change zero to very small ...
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How to sample from $c^a d^{a-1} / \Gamma(a)$?

I want to sample according to a density $$ f(a) \propto \frac{c^a d^{a-1}}{\Gamma(a)} 1_{(1,\infty)}(a) $$ where $c$ and $d$ are strictly positive. (Motivation: This could be useful for Gibbs ...
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What are the mean and variance for the Gamma distribution?

There are two forms for the Gamma distribution, each with different definitions for the shape and scale parameters. Rather than asking what the form is used for the gsl_ran_gamma implementation, it's ...
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Kullback–Leibler divergence between two gamma distributions

Choosing to parameterize the gamma distribution $\Gamma(b,c)$ by the pdf $g(x;b,c) = \frac{1}{\Gamma(c)}\frac{x^{c-1}}{b^c}e^{-x/b}$ The Kullback-Leibler divergence between $\Gamma(b_q,c_q)$ and $\...
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How to derive Poisson distribution from gamma distribution?

Let $T_1, T_2, \dots$ be iid sequence of exponential random variables with parameter $\lambda$. The sum $S_n = T_1 + T_2 + \dots + T_n$ is a Gamma distribution. Now as I understand the Poisson ...
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Conditional expection of gamma distribution on sum

I have a process which consists of a number of events and what is known is the timings between the events. What I'm trying to determine is a distribution that allows me to determine a likelyhood that ...
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How to quickly sample X if exp(X) ~ Gamma?

I have a simple sampling problem, where my inner loop looks like: v = sample_gamma(k, a) where sample_gamma samples from the ...
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1answer
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What is the expected value of modified Dirichlet distribution? (integration problem)

It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If: $ X_i \sim \text{Gamma}(\alpha_i, \beta) $ Then: $ \left(\frac{X_1}{\...
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How do you calculate the expectation of $\left(\sum_{i=1}^n {X_i} \right)^2$?

If $X_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $X_i$'s are mutually independent, what is the expectation of $$ \left(\sum_{i=1}^n {X_i} \right)^2$$ in terms of $n$ ...

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