Questions tagged [gamma-distribution]

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

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184 views

Probability of an exponential random variable being greater than a gamma random variable?

Let V have exponential(a) density, and let W be independent of V with gamma(s,b) density. Find P(V>W). What I did for this problem is I integrated the conditional probability P(V>W|W)f(w)dw from w = ...
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237 views

Gamma-2 Distribution in Bayesian

According to the material I have in hand for Bayesian Econometrics, we define the pdf of a Gamma-2 distributed random variable $Z$ with parameter $\mu > 0$ and degrees of freedom $\nu > 0 $, ...
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37 views

shape and rate of the square of a variable having a gamma distribution

from this answer (Expectation of a squared Gamma) I would like to know the shape and rate parameters of a squared gamma. I struggle a bit here. ...
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117 views

Gamma likelihood with InverseGamma prior

I've got a gamma likelihood $\Gamma(\tau_c | \alpha_k, \frac{\alpha_k} {\tau_k})$ (parameterized with shape and rate) with an InverseGamma prior $IG(\tau_k|a_0, b_0)$. I know that the resulting ...
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1answer
204 views

Integrating out a gamma-distributed parameter from a Weibull distribution

I'm dealing with a variation of the three-parameter Weibull distribution where the third parameter is randomly distributed over a Gamma distribution. The cdf takes the form: $$ G(x|\gamma) = 1-\exp\...
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125 views

Let $X$ have the gamma $(r, \lambda)$ distribution. Show that $P(X \ge 2E(X)) \le (2/e)^r$

Let $X$ have the gamma $(r, \lambda)$ distribution. Show that $P(X \ge 2E(X)) \le (2/e)^r$. I do not know how to approach this. I am thinking of Chernoff Bounds, but what trips me up is how to ...
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1answer
772 views

Uniformly Most Powerful Test Gamma Distribution

In this worked-out solution, I'm convinced there is a typo: In standarizing the variable, I understand how typically, we're supposed to subtract the mean from the variable in the numerator, so why ...
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623 views

Exponential Likelihood Ratio Test

Cross posting from math.stackexchange... Suppose that $X_1, ...X_n$ is a random sample of size $n$ drawn from an exponential distribution with unknown parameter $\theta$. Suppose that it is desired ...
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49 views

Asymptotically confidence interval of level $1-2\alpha$

For fixed $k$, the variables $(2k+1)\hat{f}_k(\lambda)$ are asymptotically distributed according the Gamma distribution with shape parameter $2k+1$ and mean $(2k+1)f_X(\lambda)$. It turns out that ...
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263 views

rearranging gamma functions to solve Weibull distribution parameters

I'm trying to generate a Weibull distribution to see if it fits some data. I have an arithmetic mean and variance for this data but I'm struggling to analytically create the specific Weibull ...
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90 views

Probability square of a normal is in a range; mode of gamma

If $X \sim N(1,4)$ find $\mathbb{P}(1<X^2<9)$ If $x=2$ is the unique mode of the $X \sim \Gamma(2,\beta)$ distribution, find the parameter $\beta$. Well I tried transformation to solve ...
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460 views

Interpretation Beta coefficient regression gamma distribution

I am currently working on a panel data model of 30 companies over 10 years where the dependent variable is a score (decimal bounded between 0 and 1, continuous) while the independent are dummies and ...
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146 views

What is the problem with negative estimage for variance using optim/mle/mle2

Why fitting a correlated shared gamma frailty model in R, I obtained the negative estimate for the variance of parameters of interest. I used optim, mle and mle2 with L-BFGS-B and Nelder-Mead ...
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283 views

Is it right to use normal distribution to compute confidence intervals of a point in a zipf distribution?

I have a list of datapoints that falls within some sort of zipfian distribution or gamma distribution: ...
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1answer
133 views

Decompose mixture of gamma and gaussian distributions

I have a data, which looks like mixture of $\gamma$ and Gausian distributions: Could you help me to find parameters of these distributions?
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596 views

What's the practical meaning of alpha in a GLM with gamma family?

I am fitting several models of the form.. glm(DV ~ I(1/IV), family = Gamma(link = "log") .. and am looking for ways to compare the models obtained for different ...
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66 views

Convolve Gamma distribution with Triangle distribution?

I am working on the use of distributed delay applied to pharmacometric models. Specifically, the delay kernel I am interested in is the Gamma distribution, with non-integer shape. The historical ...
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52 views

How to compare goodness of fit between gamma mixture and beta mixture in R?

I have data presenting two peaks that I fitted with both a mixture of gamma and a mixture of beta distributions, in R. The data have for support [0,1]. How to assess which mixture (gamma vs beta) is ...
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1answer
306 views

Expected number of events from Poisson distribution with Gamma prior

The expected number of events for a Poisson distribution under a Gamma prior with parameters $\alpha$ and $\beta$ (with mean $\alpha/\beta$) is: $$ \newcommand{\paren}[1]{\left(#1\right)} \begin{...
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470 views

Hypothesis testing for Gamma distribution

I have a sample $X_1,...,X_n \sim \Gamma(\alpha, \beta)$, where $\alpha, \beta$ - unknown parameters of Gamma distribution. How to build a test for testing $H_0:\alpha=1$ against $H_1:\alpha>1$?
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2answers
155 views

Distribution of counts for event-sequence

I work with event sequence. Let's say I observe LED blinking. My sequence will look like black spikes on figure. Intervals between events distributed similarly (but not absolutely) to $\gamma$ with ...
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2k views

Deriving exponential distribution from sum of two squared normal random variables

Let $X$, $Y$ be i.i.d. random variables with distribuition $\mathcal{N}(0,1/2)$ and $Z = X^2 + Y^2$. I'd like to prove based on $X$ and $Y$ pdf's that $Z$ has exponential distribuition.
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Can I still update my prior if I have been waiting a long time without observing any successes of a Poisson process?

This is related to my previous question: How to update Poisson conjugate prior with observations of arrival time instead of counts? Using the same notation, suppose $N \sim \operatorname{Pois}(\mu)$ ...
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587 views

How to update Poisson conjugate prior with observations of arrival time instead of counts?

Suppose a random variable $N \sim \operatorname{Pois}(\mu)$, with Gamma conjugate prior such that $\mu \sim \operatorname{Gamma}(\alpha, \beta)$. Then given a sequence of $n$ observations of counts $\...
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148 views

Detect the correct distribution from a small sample size by using fitdistrplus in R

The simplest version of the issue that I am looking for help is: How to detect the correct distribution from a small sample size in R by using fitdistrplus A simpler version: I am generating some ...
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Use of Gamma Distribution for count data

I am working on my data including the insect abundance in dependence of landscape variables with a nested random effect. Since i collected the individuals in the field i have count data and thought a ...
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459 views

Is a gamma distribution adapted when the coefficient of variation is roughly the same across observations?

Each observation is a distribution of values. Computing the mean $\mu_i$ and standard deviation $\sigma_i$ for each observation $i$ showed that the coefficient of variation (CV) is roughly constant ...
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85 views

Difference of two Gamma random variables with the same scale parameters? [duplicate]

What is the distribution (probability density function) of the difference of two Gammas with the same scale? The parameters of each Gamma would be positive integers. That is, I'd like to know the ...
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2answers
105 views

Having trouble identifying this probability distribution and its parameters

For $y = \frac{1}{4}\text{ }x\text{ }e^{-x/2}$, my initial hunch was that it is a normal distribution but I wasn't able to figure out what the mean and variance would be. What is it? What are its mean ...
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56 views

Expectation of ratio between product of gaussian r.v.'s and generalized gamma r.v

Given \begin{equation}\label{eq:definition_of_z} \begin{split} \textbf{Z} = \left[\begin{array}{cccc} {z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\ {z}_{21} & {z}_{22} & \cdots & {...
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Using GLMs with gamma distribution and negative predictors

I'm currently trying to investigate relationships between habitat characteristics and animal abundances using GLMs. I've gone through the process of whittling down possible predictors and have a final ...
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1answer
738 views

What is the correct to model inverse gamma distribution [closed]

I tried to use below R code to model inverse gamma distribution (alpha=1,beta=1). However, the resulting histogram is not alike the one plotted in the wiki. Could anyone provide any hint about this? ...
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1answer
167 views

Independence of Gamma and Beta random variables with common term

Given $P$ independent and identically distributed random variables, $X_1, X_2, ..., X_P \sim \text{Gamma}(M,2c)$ how can we prove that: $$U = X_1 + X_2 + ... + X_P$$ and $$V = \frac{X_1}{X_1 + ...
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Utilising the reparameterisation trick on non-Gaussian distributions (Dirichlet)

I'm specifically looking to apply the trick to a Dirichlet distribution. Kingma and Welling (2013) briefly talk about how the trick can be applied to non-Gaussian distributions, and state that the ...
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1k views

On real-world use of gamma distributions

I typically encountered gamma distributions to model response time after a certain event. As far as my statistics goes, that is its natural place. However, in a recent piece of work of mine, I found ...
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2answers
249 views

Expectation of the ratio between Beta and Gamma random variables

Given \begin{equation}\label{eq:definition_of_z} \begin{split} \textbf{Z} = \left[\begin{array}{cccc} {z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\ {z}_{21} & {z}_{22} & \cdots & {...
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616 views

What is the distribution of the ratio between independent Beta and Gamma random variables?

What would be the distribution of the following equation: $$y = \frac{a}{(a+d)^2}$$ where $a, d$ $\sim$ $\Gamma(M,c)$. Additionally, $a$ and $d$ are independent random variables.
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851 views

How do gamma distributions add and what would that model?

Density distributions add by convolution, and the result is also a density distribution. So writing this in the time domain, w.l.o.g., the question becomes how do we take a faster gamma distribution: ...
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256 views

How to check the correctness of calculations with a gamma distribution?

I’m reading Ponomareva, Roman, and Date (2015) and trying to generate vector $P$ of the $2Ns + 3$ probability weights: $$P =\{\underbrace{p_1, p_2, \ldots, p_s, p_1, p_2, \ldots, p_s,p_1, p_2, \ldots, ...
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5k views

Is it possible to understand pareto/nbd model conceptually?

I am learning to use BTYD package that uses Pareto/NBD model to predict when will be a customer is expected to be back. However, all literature on this model is full of mathematics and there does not ...
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2k views

How is a Chi-square distribution a gamma distribution if it only has one parameter?

I know that the gamma family of distributions are a two-parameter family, but Chi-square only has one parameter. How is a Chi-square distribution a gamma distribution if it only has one parameter?
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Finding Cramer Rao Lower bound for a bivariate parameter

I am having a hard time figuring out the Cramer Rao lower bound for a random sample of size $n$ from a population with $\Gamma(p, \theta)$ with $p, \theta$ unknown. The problem doesn't say what ...
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1answer
267 views

Efficient estimator for the mean of a Gamma distribution

Let $X_1$,$X_2$,...,$X_n$ be i.i.d. according to Gamma($\alpha$,$\beta$). Denote the mean by $\mu := E[X_i] = \alpha/\beta$. Can you find an unbiased and efficient estimator for $\mu$? MLE gives ...
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1answer
66 views

Poisson distribution and minimum parameters

I am trying to work out a research problem I recently faced. I have a group of Poisson random variables and I want to find the distribution of the first sample that is equal to a specific number. In ...
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1answer
179 views

How to assess selected response distribution [closed]

I am trying to predict the impact of readmission events (continuous) and type of readmission (medical, surgical, other) on total hospital costs. Because costs are skewed by nature, I am using a Gamma ...
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1answer
327 views

mean variance relationship of the generalized Gamma

The Gamma distribution has a mean-variance power relationship of $$var(Y) = a \mu^2$$ where $a$ is a constant and $\mu$ is the mean. Is this also the case for the generalized Gamma distribution? ...
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1answer
2k views

Random Variate Generation for the Gamma Distribution [closed]

Why am I getting the following error when I try random generation for the gamma distribution? The code I am trying to run: rgamma <- (n = 500, shape>= 1, scale = 1) This is the error I get : ...
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1answer
290 views

Survivor Function vs. Hazard Function

I'm attempting to understand what the survivor and hazard functions describe under a non-traditional context. I have data comprising distances between successive points on a line ($1D$ vector): ...
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1answer
425 views

If I have three sets of continuous data which are not normally distributed, can I still compare them with ANOVA?

I have three sets of data from some experiments. I fitted each set to different distributions, and each one fits a different distribution. For example, Gamma, Weibull, and Lognormal. If I want to ...
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1answer
2k views

GLM with Gamma-Log link: How do I do Predictions?

I am rather new to regression analysis, having a completely different background. I am trying to build a model for predictions. The distribution of my dependent variable, Value (in US$), is right ...