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Questions tagged [gamma-distribution]

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

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77 views

How do I show that $Y=2\sqrt{X_1X_2}\sim$Gamma$(2p,1)$?

Suppose that $X_1\sim $Gamma$(p,1)$ and independently, $X_2\sim $Gamma$(p+1/2,1)$. Show that $Y=2\sqrt{X_1X_2}\sim$Gamma$(2p,1)$. This problem followed a section on bivariate transformations, so I ...
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1answer
26 views

MLE of shifted Gamma

Let $X = (X_1,\dots, X_n)$ and $X_1,\dots, X_n$ be i.i.d Gamma($p,a,A$) random variables where $p$ and $a$ are known. Find the MLE of $\theta =A$. We have \begin{align*} f_{\theta}(x) &=\...
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1answer
24 views

How to interpret parameters of GLM output with Gamma log link

I am having tough time interpreting the output of my GLM model with Gamma family and log link function. My dependent variable if "Total Out-of-pocket cost" and my independent variables are "Private ...
2
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0answers
33 views

Compounding a gamma distribution with another distribution to yield a gamma

I have a gamma distributed random variable $X$, with its mean $\mu$ distributed as some other function $$ X \sim \text{Gamma}(\mu,k)\\ \mu \sim P(\theta) $$ What is the distribution $P(\theta)$ such ...
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25 views

Why Are they doing exponential distributions?

With many thanks for help in why my exercise is using a Gamma distribution, I am still confused by another part. The plot: The commentary: We may suspect from the above that there is some sort of ...
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1answer
1k views

Why would they pick a gamma distribution here?

In one of the exercises for my course, we're using a Kaggle medical dataset. The exercise says: we want to model the distribution of individual charges and we also really want to be able to ...
2
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1answer
19 views

Variance for events occurring with gamma and geometric distribution

I am presented with a problem as follows: A listener is receiving messages with a wait time in between two consecutive messages that is exponentially distributed with a mean of 1 time unit. After any ...
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1answer
20 views

Distribution transformation when having same mean and variance

Is it possible to transform the distribution to Gaussian, with same mean and variance? For example, The Gamma RV. X with K=7.5 and theta=1.0. --> The Gaussian RV. X' with not changing mean and ...
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59 views

Pareto/NBD with time-varying covariates

I am trying to incorporate time-varying covariates into the Pareto/NBD model (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.597.3165&rep=rep1&type=pdf) Model assumptions start at ...
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31 views

Concave downward link function for a glm?

I seem to occasionally find datasets where the relationship between X and Y is concave downward. It seems like it should be trivial to find a link function that fits a concave downward curve, but they ...
2
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1answer
48 views

Distribution of the sample variance $S^2$ from a normal population [closed]

Let $X_1, X_2, X_3, ….., X_n$ be $N(\mu, \sigma^2)$ distributed. Then what is the distribution of $S^2$ I have already proven that if $X_i$ are $N(\mu, \sigma^2)$, then $\frac{(n-1)S^2}{\sigma^2}$ ...
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21 views

Posterior (conjugate) prior of two parameter Gamma likelihood

This question is related to a previous question on this site. Assume some data is generated from Gamma distribution $p(x\mid\alpha,\beta) \sim \operatorname{Gamma} (\alpha,\beta)$, and both parameters ...
3
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1answer
94 views

Gamma GLM - Derive prediction intervals for new x_i

In a Gamma GLM, the statistical model for each observation 𝑖 is assumed to be $Y_i \sim Gamma(shape, scale)$, where $E(Y_i) = \mu_i = f(X_i\beta)$, and $f$ is the link function. I've used MLE to ...
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1answer
94 views

Gamma Distribution for generating random numbers

I defined a gamma distribution with following parameters: shape,scale = 4.2503, 7037. This dribuation is used to generate random numbers. The random numbers will be recalculated to the x-asix value. ...
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14 views

Distribution of X/(X+Y)of gamma variables [duplicate]

Does anyone know how to calculate the distribution of $$X/(X+Y)$$ X ~ G(p,a) Y~G(p,b) G is gamma distribution
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1answer
70 views

How Gamma generalized linear model with zero dependent variable value is derived?

I understand that Gamma distribution generates only positive values. And this is reflected in R gamma family glm function which ...
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1answer
187 views

Name for a distribution between exponential and gamma?

The density $$f(s)\propto \frac{s}{s+\alpha}e^{-s},\quad s > 0$$ where $\alpha \ge 0$ is a parameter, lives between the exponential ($\alpha=0$) and $\Gamma(2,1)$ ($\alpha \to \infty$) ...
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1answer
21 views

Is it possible to calculate the scale param θ for the Gamma distribution, given shape param K and a quantile value Q?

Stats-ophils, I am running into a problem, in which I'd like to generate a Gamma distribution (in Julia) and I know the value of the quantile Q(0.9) = 130 as well as the shape parameter k=2. Is it ...
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1answer
75 views

How do I get the CDF of a gamma distribution with mean and sd?

I have the mean and standard deviation of my data, which I determined follows a gamma distribution. I don't understand the function I found online for the CDF of a gamma distribution because of the ...
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30 views

Failed to get Uniform distribution from Gamma distribution

I read in Chapter 6 in this book that $p(K)\propto 1$ is equivalent to $e^{-K}\sim Gamma(0,0)I(0,1)$ where $K$>0 and is uniform distribution, e.g., $K \sim Uniform(0,100)$; $I(a,b)$ is the ...
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What kind of regression to use with heavily skewed data?

I have data with an explanatory variable $X$ (I think I can treat this as continuous, as scores 1-100 on a certain test) and a response variable $Y$ (continuous variable, never lower than 0). Both ...
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1answer
28 views

What’s the difference between k-theta and alpha-beta parameterization for gamma distribution?

In my book “Mathematical statistics with Applications”, written by Wackerly, it’s stated that there are two methods for parameterization of gamma distribution. The first one is k-theta and the second :...
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44 views

Finding the distribution of a piecewise function of a Gamma random variable

Let random variable $X \sim \text{Gamma}(\alpha,\beta)$. I want to derive the distribution of $Y$, where: $$ Y = \left\{ \begin{array}{ll} a X - k & \quad X \geq \frac{k}{a} \...
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1answer
80 views

How to use Gamma distributions to estimate the number of failures?

I need to calculate the expected number of failures of a product within 6 years. The time until failure is said to be gamma distributed with $\alpha=2$ and $\beta=0.5$ for a mean time between ...
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41 views

inverse Gamma Distribution and LogNormal Distribution can both discribe the data, is that a coincidence?

I am observing the fluxes of source and I am trying to learn something from it's distribution of fluxes. When I histogram my data, I can perfectly describe my data with a lognormal distribution. ...
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43 views

Characterizing a distribution

I have a set of words which in a given year has a frequency of occurrence k. I can observe that these words follow frequencies k1, k2, k3,....kn in the following year. If I have some data in the form ...
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68 views

What is the difference between Gamma GLM on log output and Gamma GLM with log link function?

Here are two models (with R code to provide some context): Model 1: Take the log of the output variable $y$, then apply a Gamma GLM using the default identity link function: ...
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1answer
22 views

Relating Chi Squared and Gamma Distributions using code?

How do we relate the chi squared and gamma distributions with code? For example if we have a chi squared distribution with cdf(x, k) and we calculate ...
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105 views

Fitting Gamma Distribution in Python with scalar factor

I have done optimization to minimize error of resulted model, however, the R-squared is still low. I try to multiply a scalar to the resulted model to obtain a better fit. The scalar will be generated ...
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1answer
59 views

Model validation in R - Gamma GLMM

I'm trying to model a response variable y with respect to a nested variable x in R. First of all, I fitted a linear mixed model (LMM) as it follows: ...
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18 views

Poisson and Gamma distribution for testing randomness

In genetics I want to test whether InDel (insertion and deletion in DNA) sizes occurs with the same probability. I heard that I should gamma distribution to model it. I found ...
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0answers
100 views

Gamma Distribution CDF Intuition; Are My Parameters Correct?

I'm learning about gamma distributions and I want to make sure my reasoning is correct for an example I found here. Suppose you are fishing and you expect to get a fish once every 1/2 hour. ...
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1answer
28 views

Optimal distribution for Acceptance Rejection Sampling

For some project I have been sampling from the Gamma distribution. I have been using the exponential distribution intensively. One method I have employed is the Acceptance rejection sampling, ...
3
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1answer
65 views

Finding method of moments estimator of $\theta$ in $\Gamma(\theta,\theta)$ distribution

Please refer to the question in image I have tried to find $ E(x) $ but i ended up with $\overline x $ = $\frac{\theta + 1}{\theta} $ which statisfies no option , i also tried to find $ E(x-1)^2 ...
3
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1answer
35 views

Should I subtract lower bound from Gamma distributed data before estimating distribution parameters?

I have some real world data that reflects waiting time in a system. As it's about waiting times I assume it's Gamma distributed and visual check (histogram overlaid by a fitted Gamma PDF) shows no ...
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1answer
67 views

When a probability density function is defined to be finite?

In "Pattern recognition and machine learning" by Cristopher Bishop, Chapter 2.3.6 (pag. 100) says that The gamma distribution has a finite integral if $a>0$, and the distribution itself is ...
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23 views

interpreting Gamma regression coefficients - individuals not totalling sum of parts

I have fit a gamma regression to a dataset, and like with traditional linear regression, I would like to calculate the contribution of each independent variable across the entire model. The problem I ...
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1answer
128 views

Gamma distribution parameters estimation [closed]

I have a set of samples taken from a population distributed with a Gamma distribution, so \begin{equation} f_X(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} \end{equation} I should ...
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0answers
223 views

MLE for Beta distribution, with $\beta$ = 3

I'm trying to calculate the Maximum-Likelihood Estimator for $\alpha$, using the beta distribution with $\beta = 3$. I'm kind of stuck at the last bit. Perhaps I've made a mistake somewhere, or this ...
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21 views

How to measure trends in the frequency of events

for my master thesis i want to figure out which website visitors increased / decreased the intervals between their visits during 3 months. Eventually, I want to segment them into increased and ...
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18 views

Marginal Distribution of Hierarchal Model Normal distribution with unknown mean and precision

I am trying to use a Hierarchical model where there I have a normal distribution with random mean and precision: $$ y \sim N(\mu, \tau)\\ \mu \sim N(M, T)\\ \tau \sim Gamma(\alpha, \beta) $$ I'm ...
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1answer
100 views

How was the pdf of the generalized gamma distribution in GAMLSS reparametrized?

I am trying to bring together the definition of the Generalized Gamma distribution in R-package GAMLSS by Rigby and Stasinopoulos and the general definition of the Generalized Gamma distribution, ...
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1answer
23 views

Probability question in Mat

My teacher give me this question: Using MATLAB, generate 10000 Random Vectors of size 500 with the PDF of Gamma distribution. Find the PDF of maximum and minimum of the generated Random vectors. (Use ...
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0answers
42 views

PDF of human population weight distribution

I have a lot of data on human weight coupled with information about gender, age, country and platform (desktop or mobile user). Looking something like this: ...
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0answers
84 views

Hypothesis testing for the Gamma distribution - Rejection region

Suppose under $H_0$, a test statistic $T$ has a gamma distribution with paramaters $\theta$ and $k$. Suppose also that the distribution of $T$ under $H_1$ is unknown. What is the appropriate rejection ...
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2answers
169 views

Proving $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the expected value of standard normal variable

I'm looking to prove that $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} z^2\, dz$ (where $Z$ is a standard ...
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2answers
439 views

Which gamma regression model to use for extrapolation?

I'm looking for a regression model which would satify these requirements: My target variable follows the exponential distribution, so to my understanding I should use gamma loss function. I have ...
5
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1answer
150 views

Truncated Gamma Distribution

The Gamma distribution is the conjugate prior of Poisson distribution. What about the Truncated Gamma distribution? Is it still the conjugate prior of Poisson distribution?
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2answers
198 views

Numerically/approximately integrating over independent gamma variables

Problem Statement For a problem in biology, I am testing out a joint distribution of the form: $$ X \sim Multinomial(\frac{\theta_1}{\sum \theta_i}, ...,\frac{\theta_n}{\sum{\theta_i}}) \\ \theta_i \...
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27 views

In an MCMC model, what happens if the observed data contains negative and positive observations, but a strictly positive model (Gamma) is used?

Suppose that we observe data that range from negative to positive support. Now, suppose we use a likelihood specification that is a Gamma, or some other distribution with strict positive support. I ...