Questions tagged [gamma-distribution]

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

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671 views

clarifying exponential-gamma conjugate prior

I'm referring to page 22 of this white paper. On page 22, it says the following: given that $s_i \sim \text{Exp}(\theta), i = 1,..,c$ $\theta \sim\text{Gamma}(k, \Theta)$, Then the posterior ...
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What is the motivation for the formula for the Gamma distribution? [duplicate]

In my statistics class, it was proven that the sum of independent and identically distributed random variables distributed according to an exponential distribution follows a gamma distribution using ...
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185 views

Probability distribution for independent time to event

I am looking at waiting times between two events from multiple patients, so I'm looking at a gamma distribution. Turns out, the model is plotting out an exponential distribution, which if I was to ...
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Scale mixture of normals; t-distribution; proving marginal distribution equality [duplicate]

This is a review-type homework question that I'm very much stuck on. Basically, these are the assumptions: $\theta|\lambda = N(\mu,\frac{\sigma^2}{\lambda})$ $\lambda = \rm{Gamma}(\nu/2,\nu/2)$ $\...
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Why is regression with Gradient Tree Boosting sometimes impacted by normalization (or scaling)?

I read that normalization is not required when using gradient tree boosting (see e.g. https://stackoverflow.com/q/43359169/1551810 and https://github.com/dmlc/xgboost/issues/357). And I think I ...
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1answer
324 views

Mean of truncated gamma distribution using threshold

Given a gamma distribution with PDF: $f(x;\alpha;\beta) = \frac{x^{\alpha-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^\alpha}$ with a shape parameter $\alpha > 0$, a scale parameter $\beta > ...
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1answer
3k views

Interpretation of coefficients from glm Gamma

I am attempting to fit a model to a dataset with frequency (Hz) is the dependent variable. Using a generalized linear model based on a gamma distribution seems appropriate since the values of the ...
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52 views

Hypothesis testing for generalized (three parameter) gamma distribution

I have generalized gamma distribution with the following equation: $$ f(x) = \frac{\lambda^{a\tau}\tau x^{a\tau - 1}}{\Gamma(a)}e^ {{(x\lambda)}^\tau} $$ and log-likelihood function $$ l(a, \lambda,...
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1answer
82 views

Strange behavior of Gamma prior in a setting with binomial likelihood

I am trying to use Bayes theorem to estimate the probability of a binary event. To give a (simplified) example: Let's say our a priori guess of the probability of the event per trial is 0.3 (and the ...
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406 views

How shape parameters are connected with mean, variance, skewness and kurtosis of generalized gamma distribution

I am writing a code in python that can generate probability distribution with given mean (m), variance (v), skewness (s) and kurtosis (k). In scipy library of python, there is a function named ...
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Did I mess up the Poisson-Gamma relationship?

Let $X_1, X_2$ be independent, exponentially distributed random variables with mean 2. So $X_1+X_2=Z$ is gamma distributed with $\alpha=2$ and $\beta=1/2$. I am trying to solve the probability that $...
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The identity link function does not respect the domain of the Gamma family?

I am using using a gamma generalized linear model (GLM) with an identity link. The independent variable is the compensation of a particular group. Python's statsmodels summary is giving me a warning ...
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1answer
922 views

Relationship between the gamma distribution and Non-central chi squared distribution?

If $Y = \sum_{i=1}^N X_i^2 $, where $X_i \sim \mathcal{N}(\mu,\sigma^2)$, i.e. all $X_i$ are i.i.d gaussian random variables of same mean and variance, then what is the resultant PDF of $Y$? How the ...
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69 views

Question regarding the distribution of sum of random variables

Let $X_1, ... X_n$ be i.i.d random variables that have an exponential distribution with parameter $\theta$. So we know that $\sum X_n \sim \Gamma(n, \theta)$. This makes sense by working backward. ...
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Independence of statistics from gamma distribution

Let $X_1,...,X_n$ be a random sample from the gamma distribution $\mathrm{Gamma}\left(\alpha,\beta\right)$. Let $\bar{X}$ and $S^2$ be the sample mean and sample variance, respectively. Then prove ...
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How to handle repeated measures when fitting a gamma model with log link

I am not a statistician and wanted to see if anyone could help me with some statistical modeling. I have the total medical costs for thousands of patients (total yearly cost for each patient) along ...
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316 views

MLE of $f(x;\alpha,\theta)=\frac{e^{-x/\theta}}{\theta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}$

Let $X_{1},X_{2},X_{3},...,X_{n}$ be a random sample from a distribution with pdf $$f(x;\alpha,\theta)=\frac{e^{-x/\theta}}{\theta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}I_{(0,\infty)}(x ),\alpha,\theta&...
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1answer
65 views

Probability of the generalized gamma distribution

I am trying to compute the value of $\bar F(x)=1-F(x)$ where F(X) is the generalized Gamma distribution. I found that this distribution is also called the equilibrium distribution of Weibull. Someone ...
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Variance of 1 over sum

I was given to prove that if we're given statistics of Gamma distributed random variables $X_1,X_2...X_n$ (pdf $f_{\chi}(x,\alpha,\lambda)=\frac{{\lambda}^{\alpha}x^{\alpha-1}}{\Gamma(\alpha)}e^{-{\...
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1answer
208 views

finding quantiles of a kernel density estimation

I used R to find kernel density estimates of my dataset (for experiment I used 1000 samples generated from a known distribution in this step). I used code density()...
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1answer
455 views

How to improve fit of distribution to data

I'm trying to fit one of common expenential distributions to data using histfit. However it seems that results aren't as good as expected - it seems that peak should be higher. Histograms presents ...
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1answer
3k views

Why do we use inverse Gamma as prior on variance, when empirical variance is Gamma (chi square)

Let $$X_i\sim \mathcal{N}(0,\sigma^2)$$ than we know that $$\sum_{i=1}^N\frac{X_i^2}{N}\sim\Gamma(\frac{N}{2},\frac{2\sigma^2}{N})$$ that the empirical variance follows a Gamma distribution. How do ...
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212 views

Power of a random variable gamma distributed

If a random variable $X$, follows a Gamma distribution. I know that the distribution is given by: $$F_X(x)=\frac{1}{\Gamma(\alpha)}\int_{0}^{x/\theta}u^{\alpha-1}e^{-u}du$$ Now, If I need to consider ...
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171 views

how to get posterior distribution of beta with gamma prior

I have $X_1, ..., X_n \sim beta(\theta,1)$ and $\theta \sim gamma(r, \lambda)$ and wish to compute the posterior distribution. Since $f(\textbf{X} | \theta) = \theta^nx^{n(\theta-1)}$ and $\pi(\...
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240 views

expectation of Log Noncentral Chi

Let $X$ follow non central chi distribution, the formula of $\mathbb{E}[\log(X)]$ (or equivalently $\mathbb{E}[\log(X^2)]$ where $X^2$ is Non central chi square) can be found here and here. However, ...
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Quotient of Pareto and Gamma random variables

I cannot find an explicit formula for the quotient of a Pareto random variable divided by a Gamma random variable. The only that I found is something like, for $P(X)$ pareto's like and $P(Y)$ Gamma's ...
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394 views

Generalized Gamma: log normal as limiting special case

The Generalized Gamma Distribution has p.d.f. defined as follows (see e.g. here for reference): $$ f(t; \theta, k, \beta)=\frac{\beta }{\Gamma (k)\cdot \theta }{{\left( \frac{t}{\theta } \right)}^{k\...
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Is my understanding of “family of distributions” correct?

As I was looking to understand the concept of "family of distributions", I stumbled upon this answer. However, I was a bit confused with answer and I'm hoping that someone may be able to clarify for ...
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Integrate Gamma pdf with respect to shape parameter alpha

Is there a trick to integrating a Gamma pdf with respect to the shape parameter alpha? I have yet to come up with a way to do it.
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Generating Matrix-Gamma distributed random variables

Is there an established algorithm for sampling from the Matrix-Gamma distribution (https://en.wikipedia.org/wiki/Matrix_gamma_distribution)? I suspect the procedure would be similar to generating from ...
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5k views

How to find alpha and beta from a Gamma distribution?

The task: Using the method of moments model the data (sample) as a set of 20 independent observations from a Gamma(λ, k) distribution. I have found the mean and variance but unsure how to find alpha ...
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1answer
269 views

Find the normalisation constant of this gamma function

I have a gamma priors defined as $$c_j \sim Ga(a_j,b_j)$$ where $b_j$ is the rate. The component likelihoods are defined to be $$L_j(c_j;x) = c_j^{r_j} exp\bigg\{-c_j \int_0^T g_j(x(t)) \, dt\...
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1answer
612 views

What is the expected maximum value of a gamma distribution, as a function of number of samples?

I have the following situation. I have observations that fill out a gamma distribution. (At least they seem to: the distribution of values of several thousand observations looks to the eye like a ...
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1answer
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What constitutes a large KL divergence?

I have 2 gamma distributions $X_1 \sim Ga(13,1) \\ X_2 \sim Ga(3,1)$ Where we are defining our gamma distribution probability density function for a random variable $X \sim Ga(a,b)$ to be $f(X) = \...
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Compare the quality of distribution fits

I have two random variables $A$ and $B$ they are of different size. Both are well fitted as $\gamma$ distributions. My question is to find which one is more gamma like. Could You help me to solve ...
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1answer
911 views

Elastic Net for Gamma distribution

I am investigating Elastic Net method on R to build a prediction model on pricing amount. I have about 70 dummies variables and results make sense regarding variable selection, stability... However ...
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1answer
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Fixed effects gamma estimator?

Question 18.12 in Wooldridge's Econometric Analysis of Cross Section and Panel Data describes a "fixed effects gamma estimator", which appears to be analogous to a fixed effects Poisson estimator, ...
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64 views

Determining appropriate distribution to describe wind speed data

Hey, I have a dataset of hourly wind speed data and according to literature, the weibull distribution is best suited for this case, however when I fit this distribution to the data, according to the ...
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1answer
370 views

Calculate the Kullback-Leibler Divergence for these 2 Gamma distributions

I have 2 models $P \sim Ga(115,1329.914) \\ Q \sim Ga(140,650.6775)$ and I'm looking to calculate the K-L divergence of these 2. $D_{KL}(P||Q) = \int_\infty ^\infty p(x) log \frac{p(x)}{q(x)}\,dx$...
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First two moments of the ratio of the geometric mean to the arithmetic mean of Gamma random variables

Let $X_1,\ldots, X_n$ be $n$ uncorrelated random variables from a Gamma distribution with different parameters: $X_i \sim Gamma(k_i, \theta_i)$. What is the distribution of $$ U=\log \left[ \dfrac{\...
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148 views

Modeling the mean of a gamma distribution with a GLM

I am modeling a random variable as $T_i\sim\Gamma(\mu_i, \alpha_i)$, where $log(\mu_i) = X_i + ZU + \epsilon$ $\mu_i$ represents the mean of the gamma distribution and $\alpha_i$ is the shape. I'...
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1answer
55 views

Posterior latent $t_p$-distribution

I have a random vector $Y\in\Bbb{R}^p$ that is distributed as a multivariate t-distribution $t_p(\mu,\Sigma,\nu).$ I know we can see the distribution as $$Y\; \vert\; Z=z \sim\mathcal{N}(\mu,\Sigma ...
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2answers
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What are the properties of the “unfolded” gamma distribution generalization of a normal distribution?

In a prior post, I developed an "unfolded" gamma distribution generalization of a normal distribution as an example of how to relate a gamma distribution to a normal distribution. This yielded $$ \...
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1answer
510 views

What is the convolution of a normal distribution with a gamma distribution?

Is there a closed form expression for the convolution of a normal distribution (ND) with a gamma distribution (GD)? There does not seem to be a direct method of solving this convolution.
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269 views

Does inverse of a Gamma moment generating function have a known distribution?

I have come across a moment generating function for a random variable $Y$ of the following form $$M_Y(t) = \mathbb E\left[e^{tY}\right] = \left[1 - \frac t \beta\right]^k.$$ So it is basically $[M_X(t)...
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1answer
962 views

Derivation of variance of normal distribution with gamma function

So I'm reading about the derivation of the variance for normal distribution and I don't understand the following derivation with the use of gamma function. So, if I continue this derivation the ...
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1answer
282 views

How to fit gamma distribution to events not happen? [duplicate]

I am trying to fit a gamma distribution to the failure time of a kind of bulb. I have 40 data. However only half of them are actually the failure time. The result 20 are times those bulbs being used (...
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1answer
526 views

Gamma regression and offset term

I'm testing the effect of soil treatments on root and shoot length of seeds of a species. However the seeds had different sizes, which may affect root and shoot development. I have a measure of the ...
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1answer
267 views

Gamma/Poisson Posterior Distrib Given Prior:

I need to find the model over a period of length t. This is what I've done: Based on the Bayes' theorem, the relationship between the prior, the posterior, and the likelihood function is $p(\theta|x) ...
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1answer
233 views

gamma mixed model with offset and/or weights

I have some continuous positive data $Y_{ij}$ representing accumulated quantities, where $i$ denotes subject and $j$ a state. Patients are transitioning across states sequentially, but not all ...

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