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Questions tagged [gamma-distribution]

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

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10 views

glmer model convergence question

We are working with a longitudinal dataset, with three variables: WAIP, BPSRRI and group. WAIP and BPSRRI are measured repeatedly for 10 times and group refers to the group assignment of our subjects ...
5
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2answers
106 views

Independence of ratios of independent variates

If $X= x_1/(x_1+x_2)$ and $Y= (x_1+x_2)/(x_1+x_2+x_3)$ where $x_1,x_2,x_3$ independent chi-square variates with d.f $n_1,n_2,n_3$ respectively, are $X$ & $Y$ independent? I know the condition ...
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0answers
18 views

Is my data suited for an ANOVA?

I have two types of "media" (surface and sediment). I have six toxin variants "variant". All the toxins originate in the surface water, but I want to know if certain toxins are more likely to ...
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0answers
30 views

Gamma distribution as a member of exponential family

In my lecture notes I have that the distribution of a random variable $Y$ is said to be in the exponential family if it can be written as $f(y;\theta)=exp(a(y)b(\theta)+c(\theta)+d(y))$, where $a,b,c$ ...
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0answers
34 views

Gamma distribution

We consider the gamma pdf of a random variable $Y$ that is given by: $$f(y) = (s^a \Gamma (a))^{-1} y^{a-1}e^{-\frac{y}{s}},$$ where $y \geq 0$, $s$ is the scale parameter and $a$ the shape ...
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1answer
20 views

Is the convolution of independent normal and gamma also a Pearson distribution?

The answer here What is the convolution of a normal distribution with a gamma distribution? gives the pdf of convolution of normal and gamma random variables. Is there a known random variable to which ...
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1answer
50 views

Canonical link of Gamma Distribution [duplicate]

I wonder why my professor said that Gamma's canonical link is $\frac{1}{\mu}$. My thoughts are: EDIT: $\theta$ is the canonical parameter. Since $$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
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1answer
23 views

Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
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0answers
21 views

Generate an autocorrelated Gamma sample of length N

How does one simulate an autocorrelated Gamma sample of length $N$? All I found online was about generation correlated variables and not an autocorrelated sample.
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0answers
22 views

In a gamma regression, how can i interpret coefficients?

My question is pretty simple, i have done a bayesian gamma regression with an inverse link, so: $\eta_i$=$\beta_0+\beta_1x_{i1}+\dots+\beta_px_{ip}$ < using an inverse link, mu is the ...
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0answers
37 views

hypothesis testing - gamma distribution

Let W = Y/B0 be a Random variable that has a gamma(2n,1) distribution. [Y has a gamma(2n,B) distribution and W = Y/B]. i) Suppose you want to test H0 : B ≤ B0 against H1 : B > B0 for some B0 > 0. How ...
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1answer
42 views

Robust analogues of Mean, CV and Skewness

I need to characterize the mean, CV and skewness of my observed data (it is gamma-like distributed). This data is an artifact (outliers) enriched, so I decide to use robust statistics: median, ...
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0answers
107 views

Characteristic function for a Chi-squared distribution

I would like to directly derive the probability density function (PDF) for a Chi-squared distribution with $k$ degrees of freedom using characteristic functions. If $X_{1}, X_{2}, \dots, X_{k}$ are ...
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0answers
32 views

Pearson differential equation distribution: gamma distribution derivation

How do I do 30a? It becomes extremely complicated if I solve the diff eq. with all the constants involved...
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1answer
104 views

Show posterior mean can be written as a weighted average of the prior mean and MLE

Suppose $Y_1, \dots Y_n$ are exponentially distributed: $Y_i | \lambda \sim Exp(\lambda)$. Find the conjugate prior for $\lambda$, and the corresponding posterior distribution. Show that the posterior ...
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1answer
28 views

Why does not the weighted sum of gamma distribution come from weighted gamma variables?

If $Z\sim 0.3\Gamma(\alpha _1,\beta _1)+0.7\Gamma (\alpha _2,\beta_2)$, why isn't $Z=0.3X_1+0.7X_2$? $X_1\sim\Gamma(\alpha _1,\beta _1)$ and $X_2\sim\Gamma(\alpha _2,\beta _2)$?
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2answers
3k views

Do test scores really follow a normal distribution?

I've been trying to learn which distributions to use in GLMs, and I'm a little fuzzled on when to use the normal distribution. In one part of my textbook, it says that a normal distribution could be ...
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1answer
37 views

How can a sum of squared random variables distributed normally with mean $\mu$ and variance $\sigma^2$ be represented as a gamma distribution?

Firstly, I'm sorry about the formatting, I hope someone can come along and give me a hand with this. I have a sum of squares of $n$ random variables distributed normally with mean $\mu$ and variance $...
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1answer
30 views

Lower incomplete gamma function format in series representation and R [closed]

As known that the lower incomplete gamma function can be written as $\gamma(a,x) = x^{a}e^{-x}\sum_k^\infty{{x^{k}}\over a^{k+1}}.$ What is the format for $\sum_j^\infty{\gamma(v/p-j,rx^{p})} $ in ...
4
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1answer
107 views

Generalized incomplete gamma function

As I know there is a built in function for incomplete gamma function, incgam(x, a), in R. May I know is there a built in function for the generalized incomplete gamma function? Or how can I modify the ...
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1answer
26 views

Unexpected estimate in Gamma GLM summary output

I have a question. How on earth is it possible to have a negative estimate for a form of a nominal variable (two forms: "HM" and "LM") when it should be positive? I'm modelling a positive continuous ...
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1answer
59 views

Deriving marginal pdf from joint pdf

Problem setup: $X \sim \Gamma(\alpha,\beta)$, $f_{Y|X}(y|x)= \tau xy^{\tau-1}e^{-xy^\tau}$ for $y>0$ and $f_{Y|X}(y|x)=0$ for $y\leq0$, where $\tau\geq1$ is a constant. I am asked to derive the ...
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0answers
140 views

Showing that a Gamma distribution converges to a Normal distribution

Consider $G = \operatorname{Gamma}(p)$. As $p$ goes to $\infty$, the Gamma becomes more and more bell-shaped. How do I show that $\frac{G - p}{\sqrt{p}} \to Z \sim N(0,1)$ as $p \to \infty$? I ...
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0answers
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Bayesian Inference: Modeling checkout times at a store

I am currently learning how to use Bayesian inference. I have been making up problems (by defining some population parameters) and then trying to infer those values from samples. I recently made up a ...
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1answer
33 views

Gamma-gamma conjugacy for rate parameter of Gamma distribution

the question is as follows. Assume the shape $r$ is a known constant. For $x \sim$ Gamma(shape = $r$, rate = $v$), the p.d.f is: $$p(x|r,v) = x^{r-1}e^{-vx}v^r/\Gamma(r)$$ a) Show that the $\theta \...
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0answers
55 views

What is Gamma model?

Recently I saw a person's resume online and it is said he used ARMA and Gamma model to analyze time-series pattern of bond market volatility.I know what ARMA model and Gamma distribution is but not ...
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1answer
61 views

Survival time problem exponential with gamma prior

The survival times, in days, of patients diagnosed with a severe form of a terminal illness are thought to be well modelled by an exponential($\theta$) distribution. We observe the survival times ...
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0answers
20 views

Could continuous concentration data (ppb) ever be modeled with negative binomial?

We have data on compound concentrations (unit: ppb, continuous variable). There are many 0s. In fact, ~70% of the observations are equal to 0 (either true 0s or below the limit of quantification). ...
2
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1answer
38 views

How to evaluate $\int_0^\infty m^{x+1}e^{-2m}dm$ as $\Gamma(x+2)\frac{1}{2}^{x+2}$?

$\int_0^\infty \frac{m^{x+1}e^{-2m}}{\Gamma(x+1)\Gamma(2)}dm =\frac{\Gamma(x+2)\frac{1}{2}^{x+2}}{\Gamma(x+1)\Gamma(2)}$ How does the left side equal the right side? I understand that the gamma ...
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0answers
30 views

Not sure if a gamma glm or glmm is needed

I am fitting a linear model for de CO2 dataset in r, I want to predict plant uptake (always positive) using Type, conc, and treatment, a quick look at the data ...
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2answers
313 views

What is the expected value of the logarithm of Gamma distribution?

If the expected value of $\mathsf{Gamma}(\alpha, \beta)$ is $\frac{\alpha}{\beta}$, what is the expected value of $\log(\mathsf{Gamma}(\alpha, \beta))$? Can it be calculated analytically? The ...
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0answers
70 views

Which method to use when calculating the confidence interval of GLMM Gamma Regression with the lme4 package in R

I am fitting a GLMM with family gamma using the lme4 package in R. Below is a code example to simulate the gamma GLMM fitting. ...
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1answer
93 views

Is the canonical parameter (and therefore the canonical link function) for a Gamma not unique?

Consider $Y_1, \dots, Y_n$ independent from the Gamma distribution. For $y > 0$: $$\begin{align} f(y \mid \alpha, \beta) &= \dfrac{1}{\beta^{\alpha}\Gamma(\alpha)}y^{\alpha-1}e^{-y/\beta} \\ &...
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0answers
19 views

finding process corresponding to laplace transform

I have a positive stochastic process $X(t)$ with Laplace transforms $$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$ One can clearly see that the ...
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1answer
203 views

Gamma Conjugate Prior & Poisson Process

I am analyzing daily data transaction data. I am assuming that The number of transactions in every day of length t has the Poisson distribution with mean λt The number of transactions in evert ...
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0answers
18 views

glmer: distribution law for number of events / time

Sorry if this is a basic question but I can't find an answer that is clear enough, so I prefer to ask. I want to model a number of events (number of gaze) that depends on the behaviour that one ...
3
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2answers
82 views

sum of $N$ gamma distributions with $N$ being a poisson distribution

I have an event having poisson distribution with time intervals of one minute. Every event has accomplishment time with gamma distribution. I $N$ number of events start in $t$ minutes, the what will ...
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1answer
41 views

Exact Confidence Interval for Poisson using Gamma-Poisson Relationship

I'm reading Casella-Berger's Statistical Inference and trying to follow along in example 9.2.15, which constructs an exact confidence interval for a Poisson rate. In this example, the authors solve ...
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1answer
123 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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0answers
25 views

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 ...
3
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1answer
65 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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0answers
12 views

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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0answers
34 views

How to implement bayesian ab testing to gamma distributed data using pymc3

I found a simple implementation for Bernoulli distributed data of the bayesian ab tesing using pymc3 in the Probabilistic Programming and Bayesian Methods for Hackers book: http://nbviewer.jupyter.org/...
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2answers
277 views

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 ...
2
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1answer
91 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
500 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 ...
2
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0answers
39 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
46 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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0answers
181 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 ...
5
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2answers
67 views

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 $...