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Questions tagged [gamma-distribution]

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

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Finding the expectation of the Gamma density function

I am confused on how to solve this problem. I understand that there is some relationship along the lines of: $$ \Gamma(\alpha) = \int e^{(-t)}t^{\alpha -1} $$ $$ \Gamma(\alpha) = \int e^{-x/\beta}(...
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Integration to Gamma?

So my teacher is integrating to find the expectation of a marginal distribution: $$E(Y) = \int_0^\infty (y) \frac{1+y}{2} e^{-y} dy$$ and goes straight to this: $$\int_0^\infty \frac{y}{2}e^{-y}dy +\...
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394 views

Using the Central Limit Theorem to form CI from Gamma distributed random variables

Firstly, I am studying the basic concepts of statistics and so any explanations, advice and suggestions are more than appreciated. I am doing a question in my note, not sure am I doing correct or not....
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483 views

Exact distribution of MLE exponential distribution

Let $y_1, \dots,y_n$ be i.i.d. random variables from $Exp(\theta)$, where $\theta$ is scale parameter. I've found the MLE $$\hat \theta=\frac{\sum^{n}_{i=1}y_i}{n}$$ Now I need to find the exact ...
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212 views

Finding P(X<Y) of 2 random variables that are gamma distributed

X ~ Gamma ($r_1$ = 3, $\lambda_1$ = 2) Y ~ Gamma ($r_2$ = 5, $\lambda_2$ = 7) Find P(X > Y) To find this, I know that I can double integrate the joint distribution of X and Y but I wanted to know a ...
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63 views

Probability that $\operatorname{Erlang}(2,\mu _2)$ is greater than $\exp(\mu _1)$

I'm trying to work out that, given that $X\sim \exp(\mu_1)$ and $Y\sim \operatorname{Erlang}(2,\mu _2)$, what is $\mathbb{P}(X<Y)$? So far I have: $$\mathbb{P}(X<Y)=\int_0^{\infty}\mathbb{P}(X&...
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Double Integral of normal cdf and gamma pdf

I am trying to solve the following double integral for a problem that came up during my research: $$\int_0^{\infty}\int_0^q \frac{1}{q} \Phi\left(\frac{u-\mu l-kQ}{\sigma l}\right)\mathrm du f_L(l)\...
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Adding 1 minute to Gamma Distribution

I am doing a simulation, and as an input, I need to add 1 more minute to the already existing Gamma Distribution that I have (Alpha = 2.58 and Beta = 24.5). What would my new alpha and beta values be? ...
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customer service time problem

In a disc shop two employees are working. When we get inside the shop, we see that the two employees are already serving two customers (one customer for each employee), with the service time being a ...
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47 views

Name of the distribution with density $P(x) e^{-x/\theta}$, where $P$ is a polynomial with positive coefficients.

The Gamma distribution of shape $k$ and scale $\theta$ has density $\frac1{\Gamma(k)\theta(k)} x^{k-1} e^{-x/\theta}$. Consider the more general distribution with density (up to a normalizing constant)...
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Distribution of the ratio of two generalized gamma distribution with same location parameters?

X $\sim$ GG(p,d,$\theta_{1}$,$\mu$) where p is power, d is shape, $\theta_1$ is scale and $\mu$ is location parameter. Also Consider Y $\sim$ GG(p,d,$\theta_{2}$,$\mu$) where p is power, d is shape, $\...
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352 views

Probability of a Gamma distribution using binomial probabilities

Suppose we have a distribution $Y\sim Gamma(\alpha=4, \beta=7)$ Given $F(Y)=\sum_{i=\alpha}^{n}{n \choose i}y^i(1-y)^{n-i}$, where $n=\alpha+\beta+1$, I need to find $P(Y \leq .7)=F(.7)$ using ...
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357 views

Differential entropy of $\Gamma$

Let $X \sim Gamma(\alpha,\beta)$ be gamma distributed random variable with probability distribution function $$ f_{X}(x)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)},\;x>0 $$ ...
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Distribution function of Sum of IID Exponentiation Variables of Variable amount

So I'm trying to determine the distribution function of a random variable, S, give: $N \sim Geo(\frac{1}{1+\lambda}) $ $S_i \sim Exp(\mu), \forall i\in [0,N]$ $S = \Sigma^{N}_{i=0}S_i$ $S = ...
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Finding the survival and distribution function of a system.

We have a random variable $X\sim Gamma(3,c)$, so that means $f(x)=\frac{c^3}{\Gamma(3)}x^2e^{-cx} ; \ x>0$, with $c$ being appropriately selected scale parameter. We also have $P(U_2 \leq x)=x^2$ ...
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Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large.

Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large. What I have come up with so far is: Let $X=$ the sum of all $X_i$, ...
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1answer
243 views

Give necessary and sufficient conditions so the sum of random variables converges almost surely

$\{X_k\}_{k}$ are independent random variables on the probability space $(\Omega, \mathcal F, P)$ and $X_k$ has gamma density $f_k(x)$ where $f_k(x)=\dfrac{x^{a_x-1}e^{-x}}{\Gamma(a_k)}$ where $x,a_k &...
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How can I calculate a future random date/time with a probability distribution like a normal distribution - gamma distribution or similar?

I need to write some code that calculates a future random date/time (i.e. essentially a period of time), and I'm looking for an appropriate probability distribution and function I can use to transform ...
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2answers
552 views

Poisson Processes with Gamma Arrivals

I think the title is the best description I can give of my problem (but I'm not 100% sure - the problem set-up has me very confused). So, given a sequence of i.i.d. Gamma RV having parameters 3, $\...
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Gaussian distribution with Gamma variance

I am using a hierarchical Bayesian model. In one part of it, I have a normal distribution with mean zero and a variance sampled from a Gamma distribution for some hyper-parameters $a_0$ and $a_1$: $$...
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1answer
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$P(E(X))$ for multiple IID Gamma Distributions

Losses relating to a type of insurance policy follow a gamma distribution with mean $30,000$ and $\alpha = 2$. For a sample of $100$ policy claims, calculate an approximate probability that the mean ...
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How to compute $=\sum_{k=0}^{r-1}\Big(\frac{-\alpha^{k}t^{k-1}}{(k-1)!}e^{-\alpha t}+\frac{\alpha^{k+1}t^k}{k!}e^{-\alpha t}\Big)$?

When I read the derivation for finding the density function of the gamma distribution, I encountered this differentiation: $$\frac{d}{dt}\Big(1-\sum_{k=0}^{r-1}\frac{(\alpha t)^k}{k!}e^{-\alpha t}\...
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1answer
166 views

Markov chain - distribution of probability of state at generic step

Let $S$ be a finite discrete state set. Let $X(i) \in S, i = 1,2, \ldots$ be a random variable sequence. I've built-up a Markov transition matrix from a set of sequences of states. State $s_1 \in S$ ...
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3answers
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How to show that $f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$is a pdf?

I have some problems trying to prove the following problem: A continuous random variable $X$ is said to have a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$ if it has a pdf ...
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1answer
178 views

Distribution of a weighted gamma random variable

I have a gamma distributed random variable $x$ with pdf $$p_x(x)=\frac{\lambda^r}{\Gamma(r)}x^{r-1}\exp(-\lambda x),$$ where $r$ and $\lambda$ are shape and rate parameters respectively. If for ...
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1answer
455 views

Let $X|Y = y\sim\text{Poisson}(y)$ and $Y\sim\text{Gamma}(\alpha, \lambda)$. Find $f_X(x)$.

Question: Let $X|Y = y\sim\text{Poisson}(y)$ and $Y\sim\text{Gamma}(\alpha, \lambda)$. Find join density $f_{X,Y}(x,y)$ and find the probability density function of $X$ (simplify until there are no ...
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1answer
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Concerning Waiting Times for a Poisson Process and the Gamma Distribution

The Statement of the Problem A certain scientific theory supposes that mistakes in cell division occur according to a Poisson process with rate $2.5$ per year, and that an individual dies when $196$ ...
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2answers
347 views

chi squared distribution of independent normal distributions that are not standard normal

I've been working on the following problem. I'm a bit confused about some of the specifics of how to arrive at the correct answer. I hope someone here could point me in the right direction: A dart ...
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Calculate probability that container will be empty over 10min period?

Meteorologists model the volume of rain water collected over a given time interval, as a realisation of a compound stochastic process in which the arrival of raindrops is modelled as a realisation of ...
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1answer
89 views

Entropy of gamma-exponential compound distribution

Following this question, I have the PDF of a gamma-exponential compound distribution as $$f(y) = \frac{\alpha\beta^{\alpha}} {(y+\beta)^{(\alpha+1)}} $$ For my application I need the entropy of ...
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1answer
224 views

Question on gamma distribution and waiting times.

Let $X_1, X_2,\dots$ be iid. random variables each with density $xe^{-x}$ for $x > 0$ and $0$ otherwise. Let $S_0 = 0$ and $S_n = X_1 +\cdots + X_n$, and $N(t) = \max\ \{n : S_n < t\}$. I need ...
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1answer
37 views

Unit Measure Axiom for the Gamma Distribution

I'm studying basic probability and in my lecture notes, it shows how the Gamma function results from the convolution of two exponential random variables. To introduce the gamma function it shows that ...
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1answer
359 views

Transforming Gamma into Chi-squared distribution

My question is about how to transform a Gamma distrbution in a Chi-squared one. I know that $X\sim\Gamma(\frac{\nu}{2},2$) is the same as $X\sim \chi^2 (\nu)$. Thus, for example in a case in which I ...
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2answers
3k views

Find the Laplace transform of the Gamma pdf

Per wikipedia the Laplace transform of the gamma distribution is $$L_X(s) = (1+\theta s)^{-k} = \frac{\beta^\alpha}{(s+\beta)^\alpha}$$ As an exercise I would like to show this.The definition I have ...
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Which parameter should be considered as “scale” parameter for Gamma distribution?

I originally posted this question on crossvalidated. In case it would be considered too "nerdy" or useless there, I also posted it here with the hope to get more replies. From Wikipedia and probably ...
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Sum of $N$ Gamma distributed random variables being $N$ a Gamma distribution random variable

Thanks in advance. Let $X$ a gamma-distributed random variable having scale $θ$ and shape $k$: $$ X \sim \Gamma(k, \theta) \equiv \textrm{Gamma}(k, \theta) $$ with its probability density function ...
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1answer
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Distribution of weighted sum of Bernoulli RVs

Let $x_1,...,x_m$ be drawn from independent Bernoulli distributions with parameters $p_1,...,p_m$. I'm interested in distribution of $t=\sum_i a_ix_i,~a_i\in \mathbb{R}$ $m$ is not large so I can ...
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1answer
489 views

Joint Distribution of a Possion Gamma distribution

Let $X\sim Poisson(m)$ and $m\sim\Gamma(2,1)$ then the join probability distribution function of $X$ and $m$ is given by $$\begin{align}f(x,m)&=f(x\mid x)\;f(m)\\&=\frac{m^xe^{-m}}{x!}\cdot\...
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1answer
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Show that $2Y/\theta$ has a chi-square distribution [closed]

The question is Let $Y$ be a random variable with a Gamma distribution with parameters $\alpha > 0$ and $\theta > 0$. Show that $2Y/\theta$ has a chi-square distribution. What is the number of ...
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1answer
103 views

laplace transform of difference between two gamma independent random variables

Knowing that the laplace transform of a Gamma distribution is given by: $$F_x(s) = \frac{\beta^a}{(s + \beta)^a}$$ and that for Z = X + Y "Sum of two independent Gamma distribution random variables"...
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1answer
103 views

Maximum of two Independent Random Variables with Erlang distribution

While I am deriving the maximum of two Erlang Indepenent random variables $$ z = \begin{cases} l, & \text{if $l$>$h$ and $l$>0} \\ h, & \text{if $h$>$l$ and $l$>0} \\ h, &...
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1answer
135 views

Distribution of the difference between two random variables

I have two independent random variables of Erlang distribution or you can consider them Gamma distributions but they are positive. Z = Y-X The difference between them should be a distribution that ...
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1answer
221 views

Sum of two independent exponentially distributed random variables

Suppose that the time between calls from your aunt Debie has an exponential distribution with a mean time of 3 days. What is the probability that you will get two calls in less than 3 days? So trying ...
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Expected value of minimum of $n$ Gamma functions? [closed]

Suppose I have a set of $n$ Gamma functions, all with different parameters. I then draw one sample from each function. How can I find the expected value of the minimum of these samples?
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1answer
485 views

Compound of uniform and gamma probability distributions

I am trying to compute the distribution of a uniform distribution whose upper limit is drawn from a gamma distribution. That is, $X \sim \Gamma(\alpha,\beta)$ $Y \sim U(0,X)$ We know: $f_X(x)={\...
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1answer
802 views

Finding the probability density function of a square root of a sum

As an exam review question, I have two independently distributed random variables $X$ and $Y$ which are both $\text{Gamma} \sim (2,\frac{1}{2})$ distributed. How do I find the probability density ...
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1answer
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Method of moments estimator and the maximum likelihood estimator for a sample from Gamma distribution

Suppose we are given a random sample $x_1, \dots ,x_n$ from a Gamma distribution. I.e., $ x_i \sim G(3,\beta)$, $i = 1,\dots ,n.$ Find the method of moments estimator and the maximum likelihood ...
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812 views

Central limit theorem for positive random variables

Let $X_1, \ldots, X_n$ be a set of $n$ i.i.d. samples of a non-negative random variable $X$ with $\mathrm{E}(X)=\mu$ and $\mathrm{Var}(X)=\sigma^2$. By the central limit theorem, the sample mean $\hat{...
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2answers
134 views

Joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$ for independent exponential random variables $X_1$ and $X_2$

If $X_i$, $i=1,2$, are independent $gamma(\alpha_i,1)$ random variables. Find the joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$. I am trying to use transformation method to solve it. Let ...
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2answers
550 views

If $1\leq \alpha$ show show that the gamma density has a maximum at $\frac{\alpha - 1}{\lambda} $

So using this form of the gamma density function: $$g(t) = \frac{\lambda^{\alpha}t^{\alpha -1}e^{-\lambda t}}{\int_0^\infty t^{\alpha - 1} e^{-t} \, dt} $$ I would like to maximize this. Now i was ...