Questions tagged [gamma-distribution]

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

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Why does the unbiased statistic in this example be MVUE immediately?

I am reading "Introduction to Mathematical Statistics" edition 8 by Robert V. Hogg et al. to familiarize myself with sufficient statistics. I got stuck by an example in Sec. 7.3 of the book. ...
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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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Expected time until k failures among n items each independent and having exponential distribution

This is the question I am trying to solve. "Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution ...
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Renewal process with inter-arrival time distributed as gamma: Model estimation

Let's start with the Poisson process: If $N_t$ is a Poisson process with parameter $\lambda$, then we know that the inter-arrival time distribution is an exponential distribution with parameter $\...
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How to find p-value when null distribution is of gamma and given observation

Doing a hypothesis question, where $$H_0: \lambda=10$$ $$H_1: \lambda \neq 10$$ null distribution to be ~$\gamma(\alpha=20,\lambda=10)$, where 10 is in rate, and an observed sample =0.8, how do I ...
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If $X\sim G(a,b_{1})$ and $Y\sim G(a,b_{2})$, then what will be the density function for U=min(X,X+Y)?

Let $X$ and $Y$ two independent random variables for gamma distributions with common shape parameter $a$ and different rate parameter $b_{1}$ and $b_{2}.$ If $U=\min(X,X+Y),$ then what will be the ...
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MLE of the Gamma Distribution

The positive random variables $X_{1}, X_{2},...X_{n}$ are independent observations having the Gamma distribution $Ga(3,\frac{1}{\eta})$, with density function: $\frac{x^{2}}{2\eta^{3}}e^{\frac{-x}{\...
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Sum of $r$ independent gamma random variables - p.d.f. technique.

Let $X_1, X_2, \ldots, X_r$ be $r$ independent gamma variables with parameters $\alpha = \alpha_i $ and $\beta = 1$, $i = 1, 2, \ldots, r$, respectively. Show that $Y_1 = X_1 + X_2 + \cdots + X_r$ ...
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Find bounds of a gamma distribution given the area

How do I find the bounds of a gamma distribution given the area under the distribution? I am trying to find the exact 100(1-p)% confidence region for my parameter estimator and I ended up with $P(c \...
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Deriving a mixed distribution from exponential and inverse gamma

Question You are given the following: the amount of an individual loss in the year $2022$ follows an exponential distribution with mean $15000$ Between $2022$ and $2025$, losses will be multiplied by ...
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Why does Gamma distribution appear in other distributions [closed]

Why does gamma distribution appear a lot? Gamma distribution appears in Chi-square distribution, Erlang distribution. It also appears in the expected value $(E[X])$ of Pareto distribution and Weibull ...
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distribution of a transformed gamma random variable

I have $X=Gamma(a,b)$ and $Y=cX$ where c is a positive constant; I need to find the distribution of Y using the moment generating function method. I know $m_Y(t) = E e^{tY} = E e^{tcX}$ If i ...
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Sum of two independent inverse gamma random variables?

Let $X$ and $Y$ be two iid inverse gamma random variables. That is, let $X \sim \mbox{Inverse Gamma}(\mbox{Shape Parameter} = \alpha, \mbox{Scale Parameter} = \beta)$ and $Y \sim \mbox{Inverse Gamma}(\...
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Defining a chi-square distribution with zero degrees of freedom

Give a reasonable definition of a chi-square distribution with zero degrees of freedom. My textbook offers a hint to use the moment generating function (m.g.f.) of a distribution that is $\mathcal X^...
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Gamma distribution - Poisson property

From my textbook after discussing the Gamma distribution and using the Poisson process postulates: It is left as an exercise to verify that $$ \int_{\lambda w}^{\infty}\frac{z^{k-1}e^{-z}}{(k-1)!} \ ...
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Conditional posterior distributions for Poisson-Gamma model

I am self-learning Bayesian statistics with the book Computational Bayesian Statistics by Turkman et al. and I am currently stuck on Problem 6.5: We consider a hierarchical event rate model with a ...
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Prove that the MLE of the lambda parameter of Gamma Distribution is unbiased

Apologies for using screenshots, I just don't have time to remember and look up all of the formatting tips. Alpha is known. Taking the expected value of the MLE I get : My first question is what am ...
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How to convert Gamma distribution into Chi square distribution?

I have to find the distribution of the r.v. $-\sum_i^{10} log(x_i)$ and present it in the form of $\chi^2$ distribution. Given that: $-log(x_i) \sim Exp(\theta) $. Then I know that the sum of ...
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The expectation of arrivals for Gamma renewal process

Suppose that $N_t$ is a renewal process with the assumption that the lifetime distribution (i.e., the interarrival distribution) is a gamma distribution with parameters $\nu$ and $\lambda$. Then, I ...
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Why doesn't the gamma distribution have the memoryless property?

The gamma distribution essentially tells us the probability of $k$ events happening in a given amount of time, $t$. It seems to me that there are certain examples of the gamma distribution where it ...
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Is this proportional to a Gamma distribution?

I have this distribution for $\theta$ $$\gamma \frac{1}{\theta^{n+2}} \exp\left\{-\frac{1}{\theta} \left(\frac12\sum_{i=1}^nX_i^2+a\right)\right\},$$ where $\gamma$ is a normalization factor ...
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Poisson Arrival time and Gamma distribution

I have a question on poisson arrival times / gamma distribution that I just cannot understand as of now. Question: The times when goals are scored in a 60 minute match are modeled as a Poisson process....
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Deriving an exact confidence interval for parameter of an exponential random variable

Problem Let $X \sim exp(\theta)$ and $X_1,..,X_n$ a random sample from $X$. I've computed the MLE and the Fisher information number : \begin{align*} \theta_{MLE} &= \frac{1}{\bar{X}} \\ I(\theta) &...
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Approximate Normal Distribution on Gamma Distribution

I have a question on approximate normal distribution on gamma distribution. Below is my task: Suppose that requests to a web server follow the Poisson model with rate $$λ=\frac{1}{3.89}s$$ With a ...
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Show that expression is Gamma distribution with given parameters

I had the expression $$\frac{1}{p(y)}(n\lambda)^k e^{-n \lambda}$$ for $k \in \{0,1,2,...\}$ where we here have that $k=\sum_{i=1}^n y_i$ where I think I can ignore $p(y)$, but if not it is given by $...
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Bayesian statistics: show that corresponding posterior is a proper distribution

We assume that $y_1,...,y_n$ are outcomes of iid. random variables $Y_1,...,Y_n$ with $Y_i \sim poiss(\lambda)$ Now we consider a Bayesian approach. I have found the likelihood to $Pr[\bar{y}=k/n]=Pr[...
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Connection between the Gamma function and gamma distribution

The Gamma function is a generalization of the factorial ($\Gamma(n)=(n-1)!$). It is also the normalizing constant for the Gamma distribution and this happens to be (when its rate parameter is $1$) the ...
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Suppose that a random variable $X$ is distributed according to a gamma distribution with parameters $\alpha = 6$ and $\beta = 2$. Find these values.

Suppose that a random variable $X$ is distributed according to a gamma distribution with parameters $\alpha = 6$ and $\beta = 2$, i.e., $X \sim \text{Gamma}(6, 2)$. A.) Computer the mean and variance ...
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Gamma distribution of maximal variance

The Gamma$(\alpha,\beta)$ distribution has pdf $f(x) = x^{\alpha-1}e^{-\beta x}$, with $E(X)=\frac{\alpha}{\beta}, Var(X) = \frac{\alpha}{\beta^2}$. Find gamma distribution of maximal variance subject ...
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Distributions with a specific property

I'm looking for examples of distributions for which $\frac{1 - F(t + \Delta t)}{1 - F(t)} \to 0$ as $t \to + \infty$, $\forall \Delta t >0$. Here $F$ is the cdf. It looks like exponential ...
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Power function for a Uniformly Most Powerful Test

Question Suppose I am interested in the following problem : \begin{align} H_0 : \ \theta \geq \theta_0 \ \text{versus} \ H_1 : \ \theta < \theta_0 \end{align} If $X \sim B(\theta,1)$, $-\sum_{i=...
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What is the distribution of sum of $\ln(X_i)$ if $X_1, ..., X_n$ are from Gamma distribution?

Let $X = (X_1, ... X_n)$ where $X_i$ are iid from distribution gamma with parameter $p$ in the following form: $$f_p(x) = \frac{\lambda^p}{\Gamma(p)}x^{p-1}\exp[-\lambda x]$$ for $x > 0$ and $0$ ...
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Posterior distribution following gamma distribution

I have a gamma-distribution T with known parameters $\alpha$ and $\beta$, and a random sample with $N=10$ from a gamma distribution with known $\alpha$ and $\beta$ equal to $1/T$. I'm trying to find ...
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UMP unbiased test

I was wondering if any of you folks could help me with this statistics problem. Here is the problem : set T an exhaustive statistic which $T \sim \Gamma(n,1)$. We need to find a UMP unbiased test for ...
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Derivation of a conditional probability mass function which involves Geometric and Gamma random variables

In the $6$th chapter of the textbook A first course in probability by Ross, there is the following question: Let $N$ be a geometric random variable with parameter $p.$ Suppose that the conditional ...
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We have an iid $X_i=(X_1,..,X_n)$ from distribution $f(x) =(\theta /2)\exp (−\theta |x|)$ show that $\theta$s conjugate prior is a gamma distribution

Suppose that we have an iid sample $X_{1:n} = (X_1, X_2, ..., X_n)$ from a Laplace distribution with density of $X_i$ given by $f(x) = \frac{θ} {2}\exp (−θ|x|)$ for $x ∈ R$ and $θ > 0$ . Show that ...
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Lévy-Khintchine canonical representaion of Gamma distribution

what is the Lévy-Khintchine representation of an infinitely divisible characteristic function(ch.f.). such as Gamma distribution (ch.f. $$f(x)=(1-\frac{ix}{\lambda})^r$$) I wonder if there is any good ...
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Relation of the min of several exponential distributions, and the min of several gamma distributions

Note that I have taken a Probability Theory class (but only briefly mentioned that gamma distribution is the sum of exponential distributions) before, so somewhat rigorous explanations are fine for me....
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The joint probability exponential and gamma distribution. [closed]

Assume that $X_{1}, \ldots, X_{n}$ are independent and identically distributed exponential with parameter $\lambda>0$. Furthermore, let $Y=\sum_{i=1}^{n} X_{i}$ follows as $Gamma(n,\lambda)$ ...
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The Gamma function: does $0\cdot\infty=0$?

My course notes (Mathematics BSc, second year Statistics module, unpublished) have For $n=1,2,\dots$, we have $\Gamma(n)=(n-1)!$ Proof. Note that $$\Gamma(1)=\int_0^\infty e^{-u}=[-e^{-u}]_0^\infty=...
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The $q$'th moment of gamma distribution?

Let $X \sim \Gamma(\beta,\lambda)$ where $\beta>0$ is the rate paramter and $\lambda>0$ is the shape parameter. When I want to compute the $q$'th moment I get that \begin{align*} \mathbb{E}[X^q] ...
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Sum of i.i.d. random variables

I am trying to obtain the distribution of $Y = \sum_{i=1}^{N} X_{i}$ when the distribution of each i.i.d. random variable $X_{i}$ is given by \begin{align} f_{X_{i}}(x) = \begin{cases}\lambda e^{\...
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Baysian analysis of the Poisson distribution

$ D = (x_i )_{i=1:n}$ is the training data, where $x_i$ follows a Poisson distribution of parameter $\lambda$. The likelihood is $ p(D | \lambda) = \prod_{i=1}^n exp(-\lambda) \lambda^{x_i}/{x_i!} $ ...
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How to solve such complicated definite double integration

I am trying to solve this double integration, but not getting it exactly. $$P_{int} = \text{Pr}(X_{is}<X_{ie})$$ $$P_{int} = \int \int_{x_{is}<{x_{ie}}}f_{X_{ie}}(x_{ie})f_{X_{is}}(x_{is})dx_{ie}...
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Rate of convergence for gamma sums

It is well known that if $f:[0,1]\to\mathbb{R}$ is continuous then $$ \begin{align} B_n(x) &=\sum_{\nu=0}^n f(\nu/n) \binom{n}{\nu}x^\nu (1-x)^{n-\nu} \end{align} $$ converges uniformly to $f$ as $...
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Expectation for a ratio of sum of $\Gamma$ distributions

1) Introduction : I am interested in computing the variance of an observable $$ O=\frac{\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}}{\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell}\left(a_{\ell m}^{...
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