Questions tagged [gamma-distribution]

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

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1answer
74 views

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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23 views

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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1answer
71 views

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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2answers
120 views

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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84 views

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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1answer
35 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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1answer
42 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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75 views

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 ...
2
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1answer
55 views

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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1answer
106 views

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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36 views

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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1answer
85 views

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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1answer
68 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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149 views

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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1answer
198 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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1answer
113 views

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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172 views

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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2answers
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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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1answer
50 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
517 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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40 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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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
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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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1answer
24 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
27 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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90 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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66 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 ...
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1answer
57 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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35 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 ...
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2answers
337 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
608 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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25 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
290 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
208 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
28 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 ...
2
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1answer
495 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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38 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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60 views

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
84 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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50 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} \...
3
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1answer
111 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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135 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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0answers
71 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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94 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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