# Questions tagged [gamma-distribution]

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

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### Comparing generalized linear mixed models (varying the distribution & link function)

I have some questions on performing mixed models on multi-rater data when residuals are heteroskedastic. I've found some of the info on 冠通棋牌-【官网首页】 confusing and quite technical-- would be very ...

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### How do you compute ICC in a gamma loglink glmm?

I have fitted a gamma generalised linear model with a log link in r using glmer from the package lmer. The goal is to make prediction models for age in old trees for conservation purposes. The choice ...

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### What is the expected value of x log(x) of the gamma distribution?

Let $w(x) = x \log{x}$
$x \sim Gamma(\alpha = 3.7, \lambda = 1)$
Find $E[w(x)]$
I have set up the following integral:
$\int_0^{\infty} x\log{x} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha -1}...

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### Conditional distributions derivation

I am trying to solve the following problem:
Given $n$ independent observations $Y_i$ from a Normal$(\theta, \tau^{-1})$ distribution with unknown mean $\theta$ and unknown precision $\tau$, i.e
$$...

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### Score Function and Observed Information Matrix for Gamma GLM

I'm studying the theory behind fitting a Gamma GLM to data.
I've been trying to replicate the steps that programming languages (e.g. R) might take to fit this GLM to data.
The Gamma distribution, ...

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### Understanding NBD transaction process and Pareto dropout process plot conceptually

I am learning to use the BTYD package that uses the Pareto/NBD model to calculate CLV. However, I am struggling to understand certain plots conceptually.
For ...

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26 views

### Generate a probability distribution given min, max and mean value in python [closed]

I want to create a long-tailed probability distribution given min, max and mean value.
For eg: min=10, mean=15, max=1000
How can I create(simulate) such distribution and sample from it in python?
...

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### What to do if I find residuals (=deviance) patterns after aplying a GAMMA GLM in R?

I have activity data for 6 individuals (ID) obtained using two different formulas (RMS.X16 and ...

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### How is the gamma distribution used in the model developed by the Imperial College COVID-19 Response Team?

The Imperial College COVID-19 Response Team report mentions, "Individual infectiousness is assumed to be variable, described by a gamma distribution with mean 1 and shape parameter 0.25." With that ...

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### Rice $\sim R(\nu,\sigma)$ to Noncentral $\chi^2$

Statistics beginner here.
I have a sample data set which is Rice distributed, $R \sim R(\nu,\sigma)$. However, I'm interested in fitting $R^2$. According to Wikipedia,
If $R ā¼ Rice ā” ( Ī½ , 1 )$ ...

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### Gamma Generalized Linear Model (GLM) - Using IRLS algorithm

I'm struggling to understand how model parameters are estimated for a GLM fit to data $X$ with response $Y$ that is (for argument sake) Gamma ($\mu,\phi$) distributed.
Let's assume $X$ has $m$ rows ...

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### Density Function for Time until Second Arrival with Poisson Process

Problem Statement: The number of arrivals $N$ at a supermarket checkout counter in the time interval from $0$ to $t$ follows a Poisson distribution with mean $\lambda t.$ Let $U$ denote the time until ...

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### What is the distribution(root mean square gamma distribution)

What is the distribution when you sample from the gamma distribution and take the root mean square?
Please tell me how to prove

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### what is the simplified expression for the sum of probability of exponential distributed data? [duplicate]

Hi, I want to find the expression of P(Y1+...+Yn>U) as shown in the screeshot below but I have no idea how to do this. I'm not sure whether I should use Erlang distribution or gamma distribution. ...

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### Sum of Exponential and Gamma Distributions

I have been learning sums of distributions and understand that the sum of exponential distributions with parameter B is a gamma distribution with parameters a=1 and B.
However, I need to figure out: ...

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### Concept of a z-score for a gamma distribution

This is a somewhat general question. For the normal distribution we have the handy concept of the z-score that can be used to measure distance from the mean in a standardized way. I've encountered a ...

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### Multiplying 2 gamma functions [duplicate]

I want to multiply 2 Gamma distributions together, namely p($y_1$)p($y_2$) where
$y_1$ ~ Gamma(237,20) and $y_2$ ~ Gamma(12$n_0$+113, $n_0$ + 13), for $n_0 > 0$.
I ended up with ā $y_1$^236 $y_2$...

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### How to estimate (possibly Erlang 2) distribution from multiple small independent samples of different size?

The problem actually pertains to neuroscience research, but can be exemplified in layman's terms using home ownership.
Special case of Gamma distribution, Erlang 1 or Erlang 2 is a good candidate of ...

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39 views

### Distribution sum of correlated normal variables squared

I'm trying to deduce which distribution my data follows and how to estimate the parameters. I have four random variables $X_i \sim N(\mu_i,\sigma_i^2)$ where the means and variances are all different. ...

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### Calculating expected value of gamma-distributed random variable [duplicate]

I am just looking for a quick check whether my reasoning is correct when calculating $E(X)$, given that $X \sim \Gamma(\alpha, \beta)$. My calculations are as follows:
\begin{align*}
\text{E}(X) &...

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### How to find input to Gamma CDF which gives specific probability

I would like a formula which allows me to input some value for a Gamma distribution random variable, and get back the total probability density up to that point.
In essence, I would like say, a ...

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### Should I use log transformed pharmacokinetic data or use GLM gamma regression with log link?

I was taught, that when we deal with data of multiplicative nature, following the log-normal distribution, like in pharmacokinetic analyses, we should log the data first to enable classic parametric ...

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### Gamma to Beta: Successes and Failure Proportions?

I have a question after reading this article on the relation between Gamma and Beta distributions and this article on the intuitive understanding of the Beta Distribution.
Question
As the Gamma ...

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### Probability Interval for Gamma Distribution with Chi squared

I currently try to transform a Gamma into a Chi-square (X2) and calculate a 95% confidence interval.
So, let's say I have a Gamma distribution of $$Gamma(4.5, 4)$$
Given the answer in
http://math....

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### How to interpret goodness of Fit of GLM with gamma?

I'm using SPSS to create a model of y (dependent variable: 0,11;0,234;0,2324) and five independent variables. I get the following results:
...

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55 views

### Correlation between heavy metals and community data

This is a preliminary study. I want to test whether the presence of heavy metals in surface water negatively affects the benthic macroinvertebrate community.
If it's the case, I'll continue the ...

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### Variance and expectation of $\frac{1}{n}\sum^n_{i=1}X^2_i$

Let $X = (X_1, . . . , X_n)$ consist of independent and identically Normal $N(0, Īø)$ random
variables, with mean $0$ and variance $Īø \gt 0$.
The Moment Estimator for $\theta$ is given by $\hat \theta ...

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252 views

### Parameter recovery (Gamma distribution and model with a random intercept)

TL:DR version - I am trying to simulate data from a gamma distribution and then fit a Generalized Linear Mixed Model (GLMM) to recover the parameters. The parameter recovery for the fixed effects is ...

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### Should I use Binomial, Poisson or Gamma distributions? With or without a log link?

I want to run a GLM to answer a few questions about differences in diet between sex and calendar year.
Questions:
Does frequency of occurrence (FO) of pieces eaten differ between sex or year?
Does ...

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### Possible to work backward from Convolution of Distributions?

So, having discovered distribution convolution, which is a method for deriving the density of a sum of individual probability distribution densities,
$$S = X_{first\_distribution} + Y_{second\...

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### Bayes - Integral to Gamma Function

When trying to understand my professors notes, I came upon this piece and donāt understand the step he took.
$\frac{(\sum x_i+2)^{n+1}}{\Gamma (n+1)} \int_0^\infty\theta^{n+1} e^{-\theta \sum x_i+2}\...

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### What are the gamma-Pareto convolutions and how have they been used?

The Pareto distributions, i.e., density functions (pdf), are types I through IV and the type II variant; the Lomax distribution. This makes for a number of possible gamma-Pareto convolutions (GPC; ...

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### Numerical approximation to quantile function for Gamma distribution

I am building a stats/probability library in python and right now I am working on the properties of the gamma distribution. I know that its quantile function (F^-1(x)) does not have a nice closed-form ...

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### What is the distribution of squared Erlang random variable?

Let $\mathbf x=[x_1, ... ,x_K]^T$, $x\sim\mathcal C\mathcal N(\mathbf 0,\sigma_x^2\mathbf I)$, I believe that the distribution of $||\mathbf x||^2=\mathbf x^{\dagger}\mathbf x$ is Erlang. Is there ...

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### Generating pairs of random variables with given covariance and gamma marginals

I have shape parameters $k_X, k_Y$ and scale parameters $\theta_X, \theta_Y$, as well as a covariance $\sigma_{XY}$.
How do I generate random variables $(X,Y)$ such that the marginals are gamma ...

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### Prove that $U_i = \frac{1}{X_i}$ has the gamma distribution

Consider a random sample $(X_1, \ldots, X_n)$ where all $X_i$ are iid r.v. with the following density function: $$f(x, \theta) = Cx^{-(p+1)} e^{-\theta/x } 1_{[0,+\infty)}(x)$$ Find the value of $C$ ...

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### Bayesian Gamma Regression Update

I'm looking for a resource that explains how to do update the coefficients for a Bayesian gamma regression using Gibbs sampling. Specifically, if
$y_i \sim Gamma(\alpha,\beta_i)$ and my data ...

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32 views

### Deriving Marginal Distribution of Poisson [duplicate]

How do you find the marginal distribution of a Poisson distribution given a gamma(a,b) prior?

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### Divide beta from a gamma distribution to get another gamma distribution?

In the textbook, there's a distribution like the following,
$S=\sum_{i=}^{200}X_i\sim Gamma(\alpha = 200, \beta)$
then the textbook define a new function $P$ obtained by diving the $\beta$, so ...

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### How do I calculate Confidence Interval for Gamma Distributed Pivotal Quantity?

I'm studying confidence intervals and then I came across the following problem:
It's said that a random variable X has Skewed Exponencial Distribution with parameters $\alpha >0$ and $v \in \...

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### Estimation of generalized gamma convolutions

How can i estimate on a data sample parameters of a generalised gamma convolution ? To be more specific, if my estimation gives me only a gamma convolution and not a generalised gamma convolution i'll ...

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### Parameter estimation under Gamma noise distribution

I have a model as follows:
$y_i = \theta x_i + \eta_i, i=1,2,...,N$
where $y_i$ and $x_i$ are known observations greater than $0$, the $\eta_i\sim Gamma(a,b), a>1$. Now, I want to obtain the ...

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### Approximating the median of a $\Gamma(\alpha,1)$ distribution with $0<\alpha<1$

Is there a good approximation (or useful bounds) for the median $\nu_\alpha$ of a $\Gamma(\alpha,1)$ distribution with $0<\alpha<1$?
I have only been able to find things like Berg & ...

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### How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$? [duplicate]

Let $X_1 \sim Gamma(\alpha_1,1)$ and $X_2 \sim Gamma(\alpha_2,1)$ be independent random variables. How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$?
By ...

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### Beta-like tail bounds for ratios of sums of i.i.d generalized gammas?

I am trying to derive a tail bound for a random variable $Z = \frac{\sum_{i=1}^a X_i}{\sum_{i=1}^a X_i + \sum_{i=1}^b Y_i}$, where $X_i, Y_i$ are i.i.d. $GGamma(1, k, 1+\epsilon)$ random variables ...

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### Fitting a gamma distribution to truncated data

I am faced with the following truncation problem:
$$
X_i \sim \Gamma(\alpha, \beta) \\
\delta_i = \chi(X_i \le \tau_i)
$$
I can observe only $(X_i, \tau_i)$ where $\delta_i = 1$ and I have no a-...

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60 views

### Poisson-Gamma conjunction - calculating posterior [duplicate]

How to calculate posterior distribution step-by-step while given:
some observed numbers of customers from the last days
that number of clients is distributed by Poisson($\lambda$) ($\lambda$ is not ...

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### GLM Using Log-gamma Distribution

My data are skewed. Using log-normal causes a strong left-skew in the residuals. Using Gamma causes a strong right-skew in the residuals. I thought to myself, why not log transform the data and then ...

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### Fitting mixture of Gamma variate functions at once (with python)

I am trying to automate the fitting of a signal composed of several Gamma variate functions with some added noise. However, I face some troubles and I do not know how to deal with it. First I do not ...

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138 views

### Sampling from Gamma Distribution using the Rejection Method

I'm having some issues working through this practice problem. I have worked through the first portion of it, and I have the solution, but I don't understand how/why the solution does two things at the ...