Questions tagged [gamma-distribution]

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

110 questions with no upvoted or accepted answers
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5
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1answer
145 views

Solving an integral equation with inverse Laplace transform

Let $\alpha,\beta,\mu>0$. I am looking for a solution, i.e. a function $g(x)$, that satisfies $$ \frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_0^\infty g(x)x^{\alpha-1}e^{-\beta x}\,\mathrm dx=\left(\...
4
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0answers
232 views

Finding Bayes estimator with inverse-gamma prior and uniform likelihood.

Consider an i.i.d. sample of data from a population given by $$Y_i|\Psi \sim \text{Unif}(0, \Psi), \quad i = 1,2,\ldots, n,$$ with $\Psi$ having a prior distribution given by $$\Psi \sim \text{Inv-}\...
4
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0answers
828 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{...
4
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0answers
2k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
3
votes
1answer
370 views

sum of two independent scaled noncentral $\chi$-squared random variables

I want to analyze or approximate a random variable that is a sum of two scaled independent non central $\chi$-squared random variables with the same degrees of freedom. For example, $$X = X_1 + a X_2$...
3
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0answers
43 views

Survival Gamma Function

I answered a question in my probability test but I think it's wrong! Could you tell me with my solution is right? The question is: Find $$\lim_{n\to\infty}\int_{2n}^\infty\frac{1}{2^n(n-1)!}t^{n-1}e^...
3
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0answers
152 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...
2
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0answers
152 views

Characteristic Function of Gamma Distributed Random Variables

I have the following characteristic function $$\sum_{m=0}^{\infty} \frac{(is)^m}{m!} \sigma_{m,k} \frac{\Gamma(\beta + m)}{\Gamma(\beta)},$$ where $i$ is the imaginary unit, $\beta>0$, $\Gamma(\...
2
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0answers
71 views

Average response/waiting time for aggregated tasks with Poisson arrival

Suppose there is a specific computation task with Poisson arrival rate $\lambda$ that could be aggregated in a way that when a task arrives and triggers a computation which lasts for $D$ seconds, if ...
2
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0answers
391 views

UMVUE of $\sqrt{a}/b$ for Gamma distribution

Suppose $(X_1,X_2,\ldots,X_n)\sim \operatorname{Gamma}(a,b)$, independent and identically distributed with pdf: $$f(x)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx},\quad x>0$$ Find the UMVUE of $\frac{\...
2
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0answers
23 views

Why is it true that $t_i$ ~ exponential($\theta$) implies $\theta t_i$~exponential($1$)$=\frac12 \chi_2^2$

Why is it true that $t_i$ ~ exponential($\theta$) means $\theta t_i$~exponential($1$). Do we treat $\theta$ as a known constant, but why? Also, we know exponential($1$)=gamma($1,1$) and $\chi_2^2$= ...
2
votes
2answers
276 views

Gamma distribution and probability less then expected value?

Let $X\sim \operatorname{Gamma}(\alpha = 7, \beta)$, then $P(X > E(X))$ is: A) 0.35 B) 0.45 C) 0.55 D) 0.65 The answer is 0.45. This is what I have so far: $E(X)=\alpha\beta$ so I ...
2
votes
1answer
826 views

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 $X_i$, ...
2
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0answers
52 views

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 ...
2
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0answers
807 views

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 ...
2
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0answers
37 views

Second partial moment of the Gamma pdf

I would like to rewrite the following integral in terms of the (incomplete) Gamma function: $$\int_r^\infty (x-r)^2f(x;k,\theta)\,dx$$ where $f(x;k,\theta)$ is the Gamma probability density function ...
1
vote
3answers
80 views

How do I evaluate the following combination of random variables? Is it martingale?

I'm about to analyse the following expression $$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$ where $Y_j$ for all $j\in \mathbb{...
1
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0answers
17 views

Sufficiency in the exponential distribution

I am trying to show that given a random sample $\{X_i\}_{i=1}^n$ where $X_i\sim exp(\lambda^{-1})$, the statistic $T(\mathbf{X})=\sum_{i=1}^n X_i$ is sufficient by using only the definition. I have ...
1
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1answer
73 views

Evaluating product of Upper Incomplete Gamma functions

I have checked several posts but couldn't find the equivalent of $\Gamma(m,a) \cdot \Gamma(m,b)$, where '$\cdot$' means multiplication. I suspect that it can be solved by applying the equivalent of ...
1
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0answers
80 views

Erlang Case of a Gamma Distribution

For part a) I get $ E(X)=\alpha\beta=\frac{n}{\lambda}. $ Thus the answer is $\frac{10}{0.5}=20$ minutes. I am not sure how to do b). Any help? The special case of the gamma distribution in which $\...
1
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1answer
50 views

Show for $\ c>0\ $ that $\ cY\sim \ \text{Gamma}(\alpha,c\beta)$

Show for any constant $\ c>0\ $ that $\ cY\sim \ \text{Gamma}(\alpha,c\beta)$ $$Y\sim\text{Gamma}(\alpha,\beta)$$ $$f_Y(y)=\frac{1}{\Gamma (\alpha)\beta^\alpha}e^{\frac{-y}{\beta}}y^{\alpha-1} ...
1
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0answers
99 views

Inverse gamma distribution general question

I am reading a paper in the genomics field (Adjusting batch effects in microarray expression data using empirical Bayes methods. from W. Evan Johnson, Cheng Li), where they try to correct for some ...
1
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0answers
784 views

Fisher information of reparametrized Gamma Distribution

I'm trying to solve the following problem: Let $X_1,...,X_n$ be iid from $\Gamma(\alpha,\beta)$ distribution with density $f(x)=\frac{1}{\Gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$....
1
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0answers
225 views

Posteriod distribution of Normal Inverse Gamma model

I want to derive the posterior distribution (without the normalizing constant) of: $$p(\mu,\sigma^2)=p(\mu| \sigma^2)p(\sigma^2)$$ with $$\mu|\sigma^2 \sim N(2,1.7^2\sigma^2) \ \ \text{and} \ \ \sigma^...
1
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0answers
978 views

Bayesian posterior with Normal inverse gamma model

I want to derive an expression for the posterior distribution (without the normalizing constant) of: $$p(\mu,\sigma^2)=p(\mu| \sigma^2)p(\sigma^2)$$ with $$\mu|\sigma^2 \sim N(2,1.7^2\sigma^2) \ \ \...
1
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0answers
34 views

Expectation of gamma kernel estimator

I read a peaper, where the author use the kernel $$ K_{\rho_b (x),b}(t)=\frac{t^{\rho_b (x)}e^{-t/b}}{b^{\rho_b (x)}\Gamma(\rho_b (x))}$$ where $$ \rho_b (x)=\left\{ \begin{array}{ll} x/b & \...
1
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0answers
30 views

Distribution of the transformation

Let $X$ and $Y$ be independent r.v. such that $X \sim N(0,\sigma^{2})$, $Y \sim \Gamma(a,b)$, $Z = \sqrt{X^{2} + Y^{2}}$. Where $\Gamma(a,b)$ denotes the gamma distribution with shape $a$ and scale $...
1
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0answers
23 views

Sum of logarithmized Dirichlet R.V.

If $$\vec x \sim \mbox{dirichlet}(\alpha \mathbf1_k ),$$ what is the distribution (or approximation of): $$\sum_{i=1}^k \log(x_i)$$. I was able to find the solution using the characteristic ...
1
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1answer
82 views

PDF of Ratio of Normal and Gamma Random Variables

Let $X \sim N(0,1)$ and $Y \sim \Gamma\left(\frac{k}{2}, \frac{1}{2}\right)$. If $X$ and $Y$ are independent, find the pdf of $$V=\frac{X}{\sqrt{Y/k}}.$$ For this problem, I introduced $U=X$, and ...
1
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1answer
47 views

Modified Mean of Gamma Distribution

Let $X,Y$ be independent random variables such that $X \sim \operatorname{Gamma}(a,b)$ and $Y\sim \operatorname{Gamma}(c,b)$. We denote $M = E_{X,Y}\left[\frac{X}{X+Y}\right]$. We know that in the ...
1
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0answers
55 views

kurtosis of a time series -

There is an excellent paper by David Warren, published in 1986 in the Journal of Hydrology, "Outflow Skewness in non-seasonal linear reservoirs with gamma-distributed inflows" (Volume 85, pp127-137; ...
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0answers
274 views

Continuous random variable Poisson distribution

A computer manufacturer produces two types of laptops, Machine A and Machine B. They offer an automatic manufacturer's warranty of 3 years when a consumer buys one of their laptops, good for two free ...
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0answers
57 views

(1st Post) Estimating the transformation of a gamma random variable with a lognormal distribution

If X is gamma(n,$\lambda$) distributed, what should $\alpha$ and $\beta$ be such that the constructed random variable $\beta*exp(\alpha X)$ is approximated by the lognormal(0,1) distribution when n is ...
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0answers
344 views

Approximating the chi square distribution using the central limit thoerem.

I need to show that the chi square distribution can be approximated through the central limit theorem to $\frac{(X-1)}{\sqrt{\frac{2}{n}}}$ ~ Normal (0,1), where X=sample mean. Knowing that $X^2=\...
1
vote
2answers
620 views

Approximate sum of gamma distributions using normal distribution

If $X_1$, $X_2$, $X_3$, and $X_4$ are gamma distributions with $\theta=2$ and $\alpha_1=3, \alpha_2=2, \alpha_3=5, \alpha_4=3$, respectively, and $Y=X_1+X_2+X_3+X_4$, I can find P(Y$\leq$25) by ...
1
vote
1answer
44 views

Finding a pdf of a Gamma Distribution when $Z=\mathrm{arccot}\,(z)$

I have been given $X \sim \Gamma(\alpha, \beta)$, so that $X$ has pdf $$f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp(−\beta x)$$ for $x > 0$ and $0$ otherwise. I have to find a pdf ...
1
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0answers
74 views

What is a vague prior for λ in Gamma(n,λ), where n is a constant not equaling 1?

I know that for λ in Poi(λ) or Exp(λ) a vague prior for λ is π(λ) ∼ Gamma (α, β) where α and β are small, I was wondering if it was the same for λ in Gamma(n,λ) since exponential distribution is a ...
1
vote
1answer
367 views

Prior for $\lambda$ in Posson($\lambda$)

How is the prior in the following question $\sim \Gamma(0.5, 0.5)$? I am relatively new to Bayesian inference and am struggling to grasp the concept of priors. Question: $X \sim Poisson(\lambda)$ ...
1
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0answers
101 views

Integral of the power of upper incomplete gamma function

Is there any hint on how to calculate, or at least attain a upper bound on, the following integral? $\int_0^\infty\left[\frac{\Gamma(n,x)}{\Gamma(n)}\right]^Kdx$. This is in fact the expectation of ...
1
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0answers
219 views

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)\...
1
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0answers
145 views

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, $\...
1
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0answers
57 views

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 $P(U_2 \leq x)=x^2$ ...
1
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0answers
51 views

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 $a_1$: $$...
1
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1answer
227 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 ...
1
vote
1answer
491 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)={\...
1
vote
1answer
2k views

Bootstrap estimation of the 95% confidence intervals for the 95% quantile for gamma distribution

I cant find any where information or algorithm how to apply in steps the bootstrap procedure to estimate the 95% confidence intervals for the 95% quantile from a random sample. Does anyone knows how ...
1
vote
1answer
637 views

Confidence interval of Inverse Gamma distribution

Let's say I have a set of value that is inverse gamma distributed, how do I compute the 95% confidence interval? Is there a formula so that I can apply to find the range of interval?
1
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0answers
710 views

distribution of sum of double exponential random variables

I want to find out whether there is a concise expression (i.e. not a convolution) for the distribution of a random variable A which is the sum of $n$ i.i.d. rv's $B_i$, which are themselves double ...
1
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0answers
77 views

poisson mixture model

I'm a doctor trying to understand Bayesian stats because I'm tired of poking needles in patients when I know that tests don't have excellent specificities and sensitivities. I need help with a ...
1
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0answers
23 views

Unknown bounded continuous distribution

Has the continuous distribution with the following probability density function in $(0,1)$ a name? $f(x;\alpha,\beta)=\frac{1}{\alpha^\beta\Gamma(\beta)}(-\log x)^{\beta-1}x^{\frac{1-\alpha}{\alpha}}$...