Questions tagged [gamma-distribution]

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

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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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When to use weibull vs lognormal vs loglogistic for survival analysis regression?

When to use weibull vs lognormal vs loglogistic vs gamma distribution vs exponential when conducting survival analysis with survival analysis regression?
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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 https://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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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 $N(0, θ)$ random variables, with mean $0$ and variance $θ \gt 0$. The Moment Estimator for $\theta$ is given by $\hat \theta ...
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198 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 don’t 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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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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31 views

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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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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111 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 ...
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Connecting interpretations of chi-squared distribution as both gamma distribution and normal distribution

According to this post I read http://www.clayford.net/statistics/deriving-the-gamma-distribution/ the gamma distribution $\text{Gamma}(\alpha,\lambda)$ is the theoretical distribution of wait ...
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Can the logarithm of the sum of random variables be decomposed into the sum of independent random variables?

Suppose $X1$ and $X2$ are two independent Gamma random variables, both follow Gamma distribution but different scales (or rates), their shape can be same or different. Let $Z$ be another random ...
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Gamma GLMM Dispersion, Random Effects, and CoV (lme4)

So I know that in glm(), with the Gamma family, one can get the dispersion parameter through the MASS package with gamma.dispersion() or can even look at the summary output as a quick estimate. How ...
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getting negative binomial from poisson and gamma

This equation is from a statistical genetics research paper. I'm struggling to understand how they get negative binomial from the integral. x_cn is poisson and q is gamma. Is there such a rule? Or is ...
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Fisher Information for Gamma

Question: Find the fisher's information for $\mathcal{G}(\alpha, \beta)$ ,$\beta$ known. Attempt: Since $\mathcal{G}(\alpha, \beta)$ ,$\beta$ known. $$f(x|\alpha) =\dfrac{x^{\alpha-1}e^{x/\beta}}...
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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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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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112 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 ...
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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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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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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 ...
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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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23 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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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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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 ...
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55 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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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 ...

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