Questions tagged [gamma-distribution]
For problems that are related to gamma-family probability distributions.
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Posterior for Pareto type II (Lomax) likelihood with a Jeffrey's prior on tail index
For Pareto Type II (Lomax) $X_i\sim\text{Lomax}(A,\alpha)$, the Jeffrey's prior is still
$\frac{\sqrt{n}}{\alpha}$
The joint is
$
\frac{
\sqrt{n} A^{n\alpha}\alpha^{n-1}
}
{
\prod _{i=1}^n {{\left(A+...
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Gamma random variable multiplied with a negative constant
If X has a gamma distribution with shape parameter a and rate parameter b, then kX with also be gamma-distributed with rate parameter $\frac{b}{k}$. But this is only true if k is a positive constant.
...
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Joint distribution of normal and inverse Gamma for full conditional distribution
Suppose
$$x|\mu_1,\mu_2\sim N(\mu_1+\mu_2,1),$$
$$\mu_i\sim N(0,\sigma^2),\ iid$$
$$\sigma^2\sim Inv-Gamma(a,b).$$
and assume the first observed $x=1.$ We want to use Gibbs to sample $x.$
I already ...
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1
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Figuring out a probability distribution
At the instant $t = 0$ a certain radioactive focus starts emitting particles. The infinitesimal probability that the focus emits a particle in the differential interval is $\lambda dt$. Let $N$ also ...
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Distribution of linear combination of exponential gamma variables
Given $X_i$ independent random variables distributed with ratio and shape as $X_i \sim \Gamma(\alpha_i,\beta)$ and a real constant vector $ N $-dimensional $w$, I want to find the distribution of $L = ...
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Finding CLT for Gamma Distribution
It seems like a lot of examples for estimating the confidence intervals of a Gamma distribution, the parameter estimation involve one variable being known. I was wondering how to find a confidence ...
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$X$ and $Y$ are independent with the same distribution and we know distribution of $X + Y$. Find distribution of $X$
We are given the following task:
$X$ and $Y$ are independent random variables and have the same distribution. $U$ and $V$ are independent and $U, V \sim \Gamma(2,1)$. $X + Y$ has the same distribution ...
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Does it converges to $0$?
I have to show that this expression zero $\frac{\log(\bar{F}(p^{-1}(x_n))}{n(1-x_n)^2} \to 0$ for $n\to \infty$, where F is the cdf of a gamma distribution with scale $k\cdot \alpha$ and rate $\frac{\...
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The distribution of 2 gamma random variables
If we have that both $X_1$ and $X_2$ are gamma(r,$\lambda$) distributed where $\lambda$ is the rate
so the density is $\Gamma(r)^{-1}\lambda^rx^{r-1}e^{-\lambda x}$.
What is the distribution of $-(...
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Estimate minimum number of spare parts
The life expectancy of a component of a machine is $20$ and its variance is $625$. If the component breaks it will be replaced instantanously by a new one. Estimate how many spare parts you need so ...
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For a standard normal vector $P\{\|x\|_2^2 > n \theta\}$ is monotonic in $n$
I want to prove that for standard normal vectors $x \in \mathbb{R}^n, x \sim \mathcal{N}(0, I_n), x' \in \mathbb{R}^{n + 1}, x' \sim \mathcal{N}(0, I_{n + 1})$:
$$P\{\|x\|_2^2 > n \theta\} \...
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$\mathbb E(e^{P(X)})$ where $X$ is gamma distributed and $P$ a polynomial.
Suppose that $X$ is gamma distributed with scale $s$ and shape $\alpha$, and that $P$ is a polynomial of degree $n$ : $$P(x)=\sum_{k=0}^n p_k x^k.$$
When it exists (e.g., when the coefficients $p_k$ ...
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A problem involving a Normal approximation to a Gamma distribution
I'm working on a course problem (unpublished),
A marketing analyst is asked to investigate the distribution of user links within a social media platform. It is well known that a few users are very ...
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Is Pareto(a,m) a subcase of Log-Gamma if m<1?
I am approximating an empirical cumulative distribution (with domain $[0,1]$) with different curves e.g. Pareto and Log-Gamma. Since the distribution has domain $[0,1]$ I need the Pareto ($Par(\alpha,...
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Derivation of Characteristic function for Variance-Gamma process
I'm trying to derive the characteristic function of a Variance-Gamma process, but I'm stuck. So, let $X_{t} = \theta G_{t} + \sigma B_{G_{t}}$, where $B_{t}$ is a Brownian motion and $G_{t}$ is a ...
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Gamma Distribution Moments Derivation
I got the following gamma function:
$\frac{B^{a}}{\Gamma(a)}x^{a-1}\ e^{-Bx}\ I_{[0,\infty)}(x)$
I would like to derive the raw moments and thus the variance but I know that I must have made a mistake ...
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Nested Hierarchical Gamma-Poisson Model
Introduction
I have been studying the following hierarchical model
$$ X_{i} \sim \begin{cases} \texttt{Po} ( \theta ), & i = 1, \dots , k; \\ \texttt{Po} ( \lambda ), & i = k+1, \dots , n, \...
user787418
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Nested Gamma Distributions
Let $a, c$ and $d$ be positive real numbers. Let us also assume that $b \sim \Gamma (c, d) $. Next, let us define random variable $X \sim \Gamma (a, b)$. Then we would like to know the probability ...
user787418
2
votes
1
answer
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Independence and Conditional Probability Density Function
Let us assume that $n \in \mathbb{N}$ and that $a_1 , a_2 , c_1 , c_2, d_1, d_2$ are positive real numbers. Let us define random variables
$$ \theta \sim \Gamma (a_1, b_1 ) , \ \lambda \sim \Gamma ( ...
user787418
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1
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Approximating $e^{-ax}-e^{-bx}$ to a gamma density function.
Is there an analytical approximation that results in the following:
$\dfrac{x^{\theta-1}.(e^{-ax}-e^{-bx})}{\Gamma({\theta}).(a^{-\theta}-b^{-\theta})}\simeq \dfrac{x^{\theta^*-1}.e^{-cx}}{\Gamma({\...
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Finding the posterior mean using gamma distribution (numerical application)
I have a normal distribution which has been parametrised so that the precision is
$$ \tau = \frac{1}{\sigma^2} $$
So my normal distribution is
$$ f(x|\mu, \tau) = \sqrt\frac{\tau}{2\pi} e^{(-\frac{1}{...
1
vote
1
answer
73
views
Summation of independent Gamma distributions?
Let $X_k$ be independent gamma distribution:
$$X_k =\frac{X}{k}$$
where $X$ is a gamma distributions on $(0, \infty)$.
How to find the distribution of the summation of $X_k$:
$$\sum_{k=0}^{+\infty} ...
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1
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Why isn't $\frac{\beta^2}{\alpha^2}\int_{0}^p x^2 f(x, \alpha, \beta) dx = F(p, \alpha+2, \beta)$?
I'm look at the following integral:
$$\int_{0}^p x^2 f(x,\alpha, \beta) dx $$
where $f(x, \alpha, \beta)$ is the pdf of the Gamma distribution is expressed as:
$$\frac{x^{\alpha-1}e^{-\beta x}\beta^\...
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votes
2
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Asymptotics of tail function of product of 2 iid gamma variables
Suppose we have an integral of this form:
$\overline F(x)=\frac{\beta^{2\alpha-1}}{\Gamma^2(\alpha)}\int_{0}^{\infty}x^{\alpha-1}e^{-\beta(\frac{x}{y}+y)}dy$, where $\beta>0, \alpha>0$ and $x>...
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Marginal Posterior Density of Poisson
Given random variables $X_{1}, X_{2}, ...$ where $X_{1} \sim Poi(\lambda)$, and $X_{j+1} \sim Poi(\lambda + \beta \mathbb{1}(X_{j} > 0))$ where $\mathbb{1}(.)$ is the indicator function, $j = 1, 2, ...
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What happens to a Gamma distribution if it is shifted?
If $X \sim \operatorname{Gamma}(\alpha, \beta)$ then the transformation $kY$ also has a gamma distribution, i.e. $Y \sim \operatorname{Gamma}(k \alpha, \beta)$. Is there any known results to what ...
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Are continuous mixtures of the gamma distribution identifiable with respect to the scale parameter?
Consider the two-parameter Gamma($\alpha$,$\beta$) distribution with PDF
$$f(x|\alpha,\beta) = \frac{\beta^\alpha x^{\alpha - 1} \exp(-\beta x)}{\Gamma(\alpha)}, \quad x>0, \alpha>0, \beta>0,$...
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Relation of distribution of $\frac{X}{X+Y}$ where $X,Y$ are iid. rv s.t exponential distribution and parameter $\alpha>0$ and $\beta'(\alpha,\alpha)$.
At first I calculate the distribution of $X/(X+Y)$. Please correct me if I am wrong. My idea is as follows:
I construct the function $f: (\mathbb{R}^2,\mathcal B^2)\to (\mathbb{R},\mathcal B),\ (x,y)\...
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Is there a way to use this relationship between gamma distributions?
This is an exercise given by my teachers. I am stuck at (1) and don't know what to do next.
Assume that:
$$ X_1,X_2,\ldots,X_n\sim Exp\left(\theta\right) $$
And that we have the prior distribution:
$$ ...
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Gamma RV Transformations
$X_1, X_2$ are independent standard normals. Define $$W_1 = X_1^2 + X_2^2$$ and $$W_2 = (X_1^2 - X_2^2)/(X_1^2 + X_2^2)$$
How can I show that $W_1$ and $W_2$ are independent using the gamma ...
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1
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How to find the posterior distribution given data and prior distribution.
Suppose that following conditions are true:
$$ X\mid\Theta=\theta\sim N(0,\theta) $$
$$ \Theta\sim InvGamma(2,1) $$
We need to find a posterior distribution: $$ f\left(\Theta\mid X\right) $$ given the ...
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Integration of density function [closed]
I am working with two random variables: $X\sim \text{Exp}(2\lambda)$ and $Y$, which has the following density function
$$f_Y(x)=\frac{1}{\pi\sqrt{x(2-x)}},$$
where $0<x<2$. I'm trying to find ...
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answer
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Prove a random variable to be a Chi-squared distribution
Here is the thing I am trying to prove:
Suppose that the random variables $X_1, X_2, \dots, X_n$ are independent, and each random variable $X_i$ has a continuous c.d.f. $F_i$. Also, let the random ...
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The summation of Gamma distribution
If ${X_1} \sim\gamma({\alpha _1},{\beta _1})$ and ${X_2} \sim\gamma({\alpha _2},{\beta _2})$, I need to prove ${X_1}+{X_2}∼Γ({\alpha _1}+{\alpha _1},{\beta _1}*{\beta _2})$ if ${X_1}$ and ${X_2}$ are ...
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Proof: U, V follow the Gamma distribution with parameters p.
The question is : Let 0<q<p, suppose U follows the gamma distribution with parameters p, V follows the beta distribution with parameters q, p-q, and U and V are independent of each other. Proof: ...
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Asymptotic variance of a consistent estimator of gamma random variables
Given $X_1,...,X_n \sim Gamma(\alpha,1/\alpha)$ random variables for some $\alpha>0$, let $\hat{\alpha}$ be a consistent estimator of the sample average $\bar{X}_{n}$ of the sample in terms of $\...
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$X_i \sim \text{Gamma}(\alpha, \beta)$ for $i=1,\dots, n$ where $\alpha, \beta >0$. Finding the pdf for the random variable $\frac{1}{n}\sum_i X_i$.
$\newcommand{\G}[1]{\text{Gamma}(#1)}
\newcommand{\a}{\alpha}
\newcommand{\b}{\beta}
\newcommand{\rd}[1]{\mathrm{d}#1}$
${\bullet \textbf{ Basics for the question:}}$
Let $X_i$ be i.i.ds (independent ...
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Should I call this distribution a Beta distribution?
Let $m, n >0$ fixed. I have this random variable $X$ whose pdf $f(x)$ is defined as:
$f(x)=\frac{\frac{x^{m - 1}}{(1+x)^{m+n}}}{B(m,n)}, x \in [1, \infty), f(x)= 0 $ elsewhere.
N.B. Yes it is ...
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Why a conjugate inverse Gamma distribution is chosen for the noise variance
I see in many papers (I cite them below), that for a measurement model contaminated with white Gaussian noise, for Bayesian approaches, the prior of the noise variance is always taken as a conjugate ...
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Relationship among Exponential, Gamma, and Normal distribution
I am studying stochastic processes where I stumbled upon the theorem that says the sum of exponential distributions is gamma distribution. However, from the central limit theorem, we know that sum of ...
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Unconditional distribuition of a normal and gamma
Im solving Gibbs sampler related questions and Im trying to find the unconditional distribuition of $\beta | \lambda \sim N(0, \lambda^{-1}\Sigma)$ knowing that $\lambda \sim Gamma(\frac{\alpha}{2}\...
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How to approxiamte the following integral related to the Modified Bessel function of the first kind?
Maybe I could approximate the following integral related to the modified Bessel function of the first kind ?
$$\mathbb{E}_{Z_{1},Z_{2},\hat{h}} \log \frac{I_{n_{\mathrm{d}}-1}\left(2 \left|\hat{h}\...
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How do software packages compute the gamma probability density for large $\alpha$?
Consider the gamma density parameterized in terms of shape $(\alpha)$ and rate $(\beta)$:
$$
f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}\mathbf 1_{x>0}.
$$
Direct computation ...
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How to solve the following double integral with the Bessel function?
Recently, I have encountered the following integral solution problem in my research. Because it involves special functions, I cannot successfully solve it in calculation.
$$\mathbb{E}_{Z_{1},Z_{2}} \...
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Why does the unbiased statistic in this example be MVUE immediately?
I am reading "Introduction to Mathematical Statistics" to familiarize myself with sufficient statistics. I got stuck by an example in Sec. 7.3 of the book. Below are page 427 and 428 that ...
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Find $\mathbb{E}[\hat{\alpha}]$. Suggest a new estimate $\check{\alpha}$ of $\alpha$ which is unbiased, such that $\mathbb{E}[\check{\alpha}]=a$
I'm currently working on the following question:
$$Define \: m=\sum_{i=1}^n log(\frac{\theta+X_i}{\theta}) \sim Pareto(n,\theta)\: and \: i.i.d$$
$$Let \: \hat{\alpha}=\frac{n}{\sum_{i=1}^n log(\frac{\...
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How do we derive the conditional distribution for a Poisson whose rate is the product of two Gamma distributed rv?
This question is motivated by Gopalan et al. "Content-based recommendations with poisson factorization." Advances in neural information processing systems 27 (2014). https://proceedings....
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With a $Gamma(2, \frac{1}{2} )$ use the CLT to prove the random variables $ \sqrt{n}( \overline{X}_{n} - 1) \rightarrow_{d} N(0, \frac{1}{2}$
I'm currently noodling through a proof as to why a Gamma distribution of $Gamma(2, \frac{1}{2} )$ converges as per the Central Limit theorem:
$$ \sqrt{n}( \overline{X}_{n} - \mu) \rightarrow_{d} N(0,...
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Alternating Renewal Process: How to calculate variance without knowing how the two distributions depend on each other
I am trying to solve a Alternating Renewal Process exercise.
The "on" state follows a exponential distribution with mean 2.
The time in the "off" state follows a gamma distribution ...
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Joint distribution of correlated gamma random variable
I'm dealing with two correlated random variables $d_1^2 = || r_1 - r_0 ||^2$ and $d_2^2 = ||r_2 - r_0||^2$, where $|| \cdot ||$ is the standard Euclidean distance ($L^2$ norm) and $r_\cdot \sim \...
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