Questions tagged [gamma-distribution]

For problems that are related to gamma-family probability distributions.

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28 views

Sum of Probability Density Function outputs is larger than 1

I have a 2 dimensional Euclidean space. Each dimension has a size of 71, so there are $71 \times 71$. The $X$ axis ranges from -35 to 35 and the $Y$ axis from 0 to 70. Using the PDF of a Normal-Gamma ...
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45 views

Bayes' theorem and law of total probability with CDFs

Suppose $X$ has Gamma(2, λ) distribution, and the conditional distribution of $Y$ given $X = x$ is uniform on $(0, x).$ Find the joint density function of $X$ and $Y,$ the marginal density function of ...
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30 views

Gamma Random Variable Confusion

I know that the gamma random variable can be thought of as an extension to the exponential random variable. The exponential random variable measures the time it takes for one event to appear, while ...
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22 views

Find solution of complex multiplication of gamma distributions.

As my main goal is to find the path, or change in $\theta$ value as $k$ changes. So I am trying to find the solution value for $\theta$. The equation that I want to solve is : given two gamma ...
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9 views

Proving that the Beta distribution can be expressed as function of gamma functions

https://www3.nd.edu/~nancy/Math30530/Info/beta.pdf I'm trying to follow the proof in the link above. On slide 4 they apply the transform $y=\frac{t}{2s} to$: $B(a,b) = \int_{-\frac{1}{2}}^{\frac{1}{2}}...
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25 views

Show that $x < {n \over 1-P_x}$ where $P_x$ represents the CDF of gamma distribution.

It is given that $$P_x = \frac{\int_0^x w^{{(n-2) \over 2}} \exp(-{w \over 2})\ dw}{\Gamma{({n \over 2})}\ 2^{{n \over 2}}}, \text{ for } x>0$$ My approach: $$P_x = 1 - \frac{\int_x^\infty w^{{(n-2)...
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24 views

Convergence in distribution - Gamma distribution/degenerate distribution

I got a gamma distribution which is defined as followed $G_{n,\lambda}=\frac{\lambda e^{-\lambda x}(\lambda x)^{n-1}}{(n-1)!}$. The paper I am currently reading says, that die $G_{n, n/t}$ converges ...
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17 views

Welch-Satterthwaite

There is a post about Satterthwaite approximation here: Proof and precise formulation of Welch-Satterthwaite equation. However what I need to know but I can't find in above post is as follow: If we ...
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13 views

Relationship between two incomplete gamma functions

Let $G(x, xb) = \frac{\Gamma(x, xb)}{\Gamma(x)}$ and $G(ax, axb) = \frac{\Gamma(ax, axb)}{\Gamma(ax)}$, where $\Gamma(., .)$ is the upper bound incomplete gamma function, $\Gamma(x) = (x-1)!$, and $a$ ...
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18 views

MLEs for rate and scale in Gamma Distribution : On Reaching the CR Lower Bound

I recently took on an excercise to find out whether the MLEs for both scale and rate parameters of the Gamma Distribution, $\theta$ and $\beta$ respectively, reach the Cramér-Rao Lower Bound. Note ...
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41 views

Determine the mean value and standard deviation with which the elevator operates per load.

The number of people $N$ entering an elevator is approximately distributed as a Poisson of mean $\lambda = 2.3$. On the other hand, the weight $W$ of a person is modeled by a Gamma distribution with ...
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57 views

If $X$ is exponential distributed with parameter $\lambda$, how is $\bar{X}/\lambda$ distributed?

In Davison A.C., Hinkley D.V. - Bootstrap methods and their application It say's $\bar{X}/\lambda$ "has the Gamma distribution with index n and unit mean". I don't understand the sentence. ...
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49 views

Is there a way to find a good/best/near-best approximation $p(x)\cdot\exp(-a\cdot x)$ ($p(x)$ is a polynomial) to a function $f(x)$?

In my work I often encounter functions with somewhat exponential decay, usually they are non-negative and non-zero everywhere and of even symmetry (so let's just assume $x\ge0$ for simplicity). I ...
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1answer
44 views

How to define an inverse gamma distribution with a fixed mode but a changeable variance for a bayesian prior?

I'm trying to define a prior distribution for a research project using bayesian estimation that's from a non-normally distributed posterior. Since it's not normally distributed I've been recommended ...
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28 views

Representing a Rayleigh distribution by Gamma distribution

Rayleigh distribution is formulated as $$P(x\mid\sigma)=\frac{x}{\sigma^2}\exp\left(-\frac{x^2}{\sigma^2}\right) \,,\tag{1}$$ where $\sigma^2$ is the variance. $Z$ is a complex variable specified as $...
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Division of two upper incomplete gamma function

Is there a simpler expression for $\frac{\gamma(aN,\ abN)}{\gamma(N,\ bN)}$ where $\gamma$ is lower incomplete gaamma function, a>1 and b > 0. Both N and a are integers. Thanks in advance.
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How is the chi-square confidence interval derived from the inverse gamma function?

I had to derive the chi-squared confidence intervals for a AR(1) red noise model generated theoretically to fit the power spectra of a time series. The shape function of the power spectra of the red ...
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16 views

Poisson arrival conditional probability

A meteorite shower is a poisson arrival with a rate of 16.6 per minute. Given that 7 meteorites were observed during the first minute, what is the expected value of the time passed until the 10'th ...
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1answer
17 views

poisson process of a machine with 2 components

I´m having trouble with a question for my Statistics class. Say a system works using 2 components, and stops working whenever one of those 2 components break. Component A fails on average 1 time every ...
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Integral of upper incomplete gamma function

Can anyone please help me with the integral below. I would like to know whether the following relation is correct? $\int\big(\frac{t}{A}\big)^{n-1}\exp\big(-\frac{t}{A}\big) dt = -A\Gamma(n, \frac{t}{...
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How to find the marginal PDF's of Bivariate Gamma Distribution

Let $X$ and $Y$ be two random variables with joint PDF $$f_{XY}(x,y)=\frac{1}{\Gamma(a)\Gamma(b)}x^{a-1}(y-x)^{b-1}e^{-y},\quad0<x<y,\quad\text{Bivariate Gamma Distribution}$$ where $a$ and $b$ ...
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What is the CDF of a linear combination of a Gamma and a Hyperbolic random variables?

I am interested in the evaluation of the risk stemming from the combination of two weather events: heavy rain and low temperature. Rainfall $R$ can be modeled as a Gamma distribution, temperature $T$ ...
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4 views

Renewal function of a gamma distribution

The density of a Gamma(2,1) is given by f(x)=xe−x. How do I compute the renewal function m(t). The book says, that the answer is m(t)=t/2+e−2t/4−14 and I don't know how to get there. I thougt about ...
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19 views

expectation of a function of random variable

I have a gamma(a,b) random variable X. Under what condition can I write $E(f(X)) = f(E(X))$? Here I have a continuous and bounded $f:(0, \infty) \to \mathbb{R}$.
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1answer
18 views

Comment on the plots of two fitted densities on a histogram

What possible comments can I draw on this following plot? It contains plot of two fitted densities. One estimating the parameters using MLE and other using MME, that I calculated from a set of data ...
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2answers
56 views

for two independent exponential distributions, the product is gamma, proof by integration

Sorry I was perhaps totally unclear. So here is the original problem Let $X,Y,Z$ be three exponential random variables independants. I want to compute $$ P( X < Y+Z ) $$ A first way to see ...
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38 views

Exact $(1-\alpha)100\%$ confidence interval for $\theta$ in $\operatorname{Gamma}(4, \frac{1}{\theta})$

I am trying to solve the following problem for exercise purposes: Let $ X_1, ..., X_n $ be independent and identically distributed random variables with the probability density function: $$ ...
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2answers
33 views

Estimator for Gamma distribution is biased or unbiased

I have $X_1,...,X_n$ which are all iid and follow a $\Gamma(\theta,\theta)$ Let $\hat{\theta}=\sqrt{\frac{1}{n} \cdot \sum X_i}$ be an estimator for $\theta$ How do I determine if $\hat{\theta}$ is ...
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40 views

Show $\prod_{i=1}^nx_{i}$ and $\sum_{i=1}^nx_{i}$ are sufficient and complete statistics for gamma distribution

$f(x_{1}, x_{2},...,x_{n}\mid \alpha\lambda)=({\frac{\lambda^\alpha}{\alpha-1!}})^{n}{(\prod_{i=1}^nx_{i})}^{\alpha-1}\exp{-\lambda\sum_{i=1}^nx_{i}}$ $\log(\alpha\lambda)=nlog{\frac{\lambda^\alpha}{\...
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1answer
23 views

What material do I need to cover to understand the gamma function?

I'm taking a course on Bayesian statistics, and my furthest understanding of math only extends to Calculus 1, logic, and elementary statistics. I'm learning about the gamma distribution (and the ...
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13 views

Expectation of 2 Gamma Distributions with different rate parameter

Given an event that happen at 2 different rates $\lambda_1$ and $\lambda_2$ simultaneously. What's the expectated time until $\alpha$ occurances in total? From what I've googled, I found that two ...
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12 views

Gamma distribution and density

I have got an easy(I hope so) question about the gamma distribution. We have got ${\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {\beta ^{\alpha }x^{\alpha -1}e^{-\beta x}}{\Gamma (\...
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32 views

Computing the distribution of the sum of independent gamma random variables

I am trying to implement the analytic expression for the distribution of the sum of independent gamma random variables using the expression given in Moschopoulos (1985). More specifically, I would ...
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3answers
34 views

Find the $E[Y]$ where Y is a summation of N i.i.d Gamma random variables

Suppose $$Y=\sum_{i=1}^N X_i,$$ where $X_i$'s are i.i.d $\operatorname{Gamma}(\alpha,\beta)$ and $N\sim \operatorname{Poisson}(\mu)$. We also assume that $N$ is independent of $X_i$'s. Find the $E[Y]$...
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40 views

Conditional Density of an Exponential Given a Sum of Exponential (Gamma)

I'm wondering about this particular problem Conditional Density of an Exponential Given Gamma I did not understand the anwser. In fact, we didn't see Basu's theorem in class, and i would like to ...
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1answer
44 views

Expected value of Gamma cumulative distribution function

I am trying to find a closed form expression of the Gini concentration ratio (G) for a mixture of $M$ Gamma distributions. Thus, I am trying to solve: \begin{equation} G=\dfrac{1}{2\mu}\int_{-\infty}^...
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53 views

Let $X$ and $Y$ be independent exponential random variables with means $\theta_1$ and $\theta_2$. What is the probability distribution of $X+Y$?

Let's try using the moment-generating functions of $X$ and $Y$ Let $\lambda_1=\frac{1}{\theta_1}$ and $\lambda_2=\frac{1}{\theta_2}$ Then $M_{X+Y}(t)=M_X(t)M_Y(t)=\frac{\lambda_1}{\lambda_1-t}\cdot\...
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48 views

Probability between two values of a gamma distribution

I am trying to solve a problem where I have a random variable X that follows a gamma distribution of $\Gamma(\theta = 5, \alpha = 4)$. I am trying to find the probability of $P(10 \leq X \leq 30)$. ...
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25 views

Finding the probability of 𝑋<25 of a Gamma Distribution

I'm given a random variable X that follows a Gamma Distribution with mean 20 and standard deviation 10. I already figured out that $\therefore \theta = 5, \alpha = 4$. I am trying to find the $P(X \...
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8 views

How to calculate the accuracy of Welch-Sattertwaite approximations for Gamma distributions with varying beta values?

I'm trying to find the accuracy of W-S approximations. I'm using MATLAB to plot the pdf of the gamma distributions. But what do I compare it against to find the accuracy? The approximations I'm ...
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1answer
106 views

Method of Moments for gamma distribution

I have data consisting of service times which I want to model with the gamma distribution. I want to use the method of moments to estimate the parameters of the gamma distribution. I get the ...
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1answer
53 views

Sum of iid gamma distributions [duplicate]

I am trying to understand the sum of equal Gamma distributions Let $X_1,...,X_n$ be iid where all $X_i$ each follow a Gamma distribution with $shape=\lambda,scale=\beta$ Is the following then true? ...
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21 views

Find $E(e^{aX})$ if $X \sim \operatorname{Gamma}(\lambda, k)$ [duplicate]

I am trying to calculate this: Let $X \sim \operatorname{Gamma}(\lambda, k)$. If $0 < a < \lambda$, find $E(e^{aX})$. I have attempted this by integration, I inserted a picture of my ...
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1answer
30 views

$\chi^2$ PDF from standard normal PDF

I was presented with the $\chi^2$ distribution as follows: Let $Z$ be a standard normal random variable $N(0, 1)$ and consider $X = Z^2$. It has the density $$f(x) = \dfrac{1}{\sqrt{2 \pi x}}e^{-x/2} ...
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1answer
32 views

Posterior of Lindley likelihood and Gamma prior

I'm doing Bayesian analysis using a $\mathrm{Lindley}(\lambda)$ likelihood and a $\mathrm{Gamma}(\alpha,\beta)$ prior. For $n$ i.i.d. data with $\mathrm{Lindley}(\lambda)$, the likelihood is: $$f(\...
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47 views

Quantiles of gamma and inverse gamma distributions

Consider the following conditions: $m,n>1$ are the $0.025$ and $0.975$ quantiles of an inverse gamma distribution with shape $\alpha$ and scale $\beta$. In this case, can we say that the $0.025$ ...
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21 views

can someone show me how this gamma distribution conditional can be derived from a joint distribution?

We have the joint distribution at the top: $p(\alpha + \kappa, r, s| \dots)$. We have the conditional distribution below $p(\alpha + \kappa|r,s, \dots)$. I'm unable to derive the conditional ...
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1answer
48 views

How to show that if $X \sim \Gamma(n/2, 1/2)$ and $Y \sim \Gamma(1/2, 1/2)$ then $X + Y \sim \Gamma((n+1)/2, 1/2)$? [duplicate]

Since $$f_X(x) = \frac{(1/2)^{n/2}x^{n/2 - 1}e^{-x/2}}{\Gamma(n/2)}$$ and $$f_Y(y) = \frac{x^{-1/2}e^{-x/2}}{\sqrt{2} \Gamma(1/2)}$$ Using the convolution formula, we get that $$f_{X} * f_Y (z) = e^{-...
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1answer
72 views

Marginal likelihood and predictive distribution for exponential likelihood with gamma prior

Let the model distribution (likelihood) be exponential, i.e. $$ p(x \mid \lambda) := \text{Exp}(\lambda) := \lambda e^{-\lambda x} $$ and the prior distribution be gamma (shape-rate-parametrization), ...
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94 views

Solving Gamma distribution word problem

A bakery sells rolls in units of a dozen.The demand X (in 1000 units) for rolls has a gamma distribution with parameters α = 3, θ = 0.5, where θ is in units of days per 1000 units of rolls. It costs 2 ...

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