Questions tagged [gamma-distribution]

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

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How to find the third moment $E[X^3]$ of Gamma Distribution?

I'm having trouble with the following question. So far I've tried to integrate $$\int_{0}^{\infty} x^3f(x) \, dx$$ but I end up with a horrible looking integral (I assume integration by parts won't ...
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1answer
382 views

chi-square distribution >> gamma(n/2)

My professor showed the transformation from chi-square to gamma(n/2), but I don't understand it. Let X be the chi-square distribution with m degrees of freedom. If Y=X/2, Y becomes gamma(n/2). What ...
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1answer
3k views

PDF of the product of two independent Gamma random variables

I am trying to find out the density of the product $XY$ of two independent Gamma random variables $X \sim \mathrm{Gamma}(k_1, \theta_1)$ and $Y \sim \mathrm{Gamma}(k_2, \theta_2)$, where $k_i$'s are ...
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2answers
48 views

Inconsistent interpretations for the second parameter of the Gamma distribution

The following question is motivated by material from pages 358-364 of Blitzstein and Hwang's Introduction to Probability. (NB: I have modified the book's notation in various places to make my ...
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1answer
875 views

Scale versus rate parameterization in exponential and gamma distributions

In some cases, I've seen the PDFs of the exponential and gamma distribution presented as: $$X\sim Exp(\lambda) \rightarrow f(x) = \lambda e^{-\lambda x}$$ $$X\sim Gamma(\alpha, \lambda) \rightarrow f(...
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26 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 ...
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2answers
99 views

Compute some expectations involving beta and gamma distributions

a) Let $X \sim {\text {Beta}}(a,b)$, with density $f(x) = \frac{x^{a-1}(1-x)^{b-1}} {B (a,b)}$. Find the expectation of $X \ln X$ and $\frac{1}{X+1}.$ b) Let $Y \sim {\text {Gamma}}(\alpha,\beta)$, ...
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374 views

Joint PDF of Gamma Distributions

Let $W_r$ denotes time taken for the r-th occurrence of the phenomenon in Poisson process $\{N_t : t \ge 0\}$ with occurrence rate $\lambda$ $$W_r = \min\{t:N_t \ge r\},\; r= 1,2,3..$$ Here I want ...
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42 views

Statistical model with $\Gamma(\alpha_i,1)$ sample

We are given statistical sample of $X=(X_1,X_2,X_3)$, where $X_i\sim\Gamma(\alpha_i,1)$ and independent. Let $Z=X_1+X_2+X_3$ and $T$ three dimensional statistic $T:=(\frac{X_1}{Z},\frac{X_2}{Z},\frac{...
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2answers
33 views

Finding the probability of $X>3$ of a Gamma

I'm given that X follows a gamma distribution and$E(X) = 8 \text{ and Var}(X)=32$ So what I tried to do was $$k\theta=8 \text{ and }k\theta^2=32 \text{ so }\theta=4,k=2$$So now I did $$\int_3^{\...
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1answer
234 views

Probability that one station becomes empty before another.

Question: There are 2 stations A and B in series having i and j customers respectively. Customers after being served at station A are routed to station B. The service time of each of the queues are ...
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1answer
85 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 ...
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1answer
231 views

Distribution of $X$ Poisson with parameter $\lambda$ if the distribution of $\lambda$ is Gamma $(2,2)$

This is from Exercise 6.15 in Canavos' Applied Probability and Statistical Methods. I cannot get my result to match answer given at the end of the book. Given a Gamma distribution with shape and ...
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236 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-}\...
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2answers
305 views

Conditional distribution of $X$ given that $X+Y=2$

I have $X$ and $Y$ independent $\Gamma(2,a)$-distributed random variables. I am trying to find conditional distribution of $X$ given that $X+Y=2$. My solution: set $U = X+Y, V = X.$ Inversion yields $...
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1answer
984 views

Poisson distribution question, solving for time until event occurs.

I've got this lengthy question that I'm really struggling with, I'd appreciate all help i can get. "The number of customers Y arriving at a walk-in shop in the first t minutes after it opens doors [i....
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1answer
48 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 ...
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1answer
110 views

What is the integral of $\frac{1}{k!}$?

I have been struggling with the following problem in probability: Assume X is a random variable with the following probability density function: $$ P(X = k) = \frac{A}{k!}, k=0,1,2,... $$ How to ...
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1answer
34 views

How to compute the joined distribution

I have to solve this exercise: D denotes the demand for a certain product. D is stochastic, its distribution depends on the price P. Assume that D | (P = p) ∼ Γ(1, p/10) and P ∼ Γ(2, 1). Derive the ...
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194 views

Is lower incomplete gamma function convex?

Is the lower incomplete gamma function convex in terms of $x$, for $x>0$ and $s>0$? $$\gamma(s, x) = x^s \, \Gamma(s) \, e^{-x}\sum_{k=0}^\infty\frac{x^k}{\Gamma(s+k+1)}$$ My answer: It is ...
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2answers
289 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 ...
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1answer
5k views

Determine the mode of the gamma distribution with parameters $\alpha$ and $\beta$

How do you determine the mode of a gamma distribution with parameters $\alpha$ and $\beta$ ? Without looking on Wikipedia.
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1answer
348 views

Probability that queue A becomes empty before queue B?

Question: Suppose, there are 2 queues A and B having i and j customers respectively. The service time of each of the queues are Exponentially distributed with parameter $\mu_a$ and $\mu_b$ (i.e., ...
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1answer
206 views

Tilted density of Gamma dist

Here is the context of the question: $X\sim \text{Gamma}(10,2)$ I want to derive the tilted density of $f$, where $f$ is the pdf of $X$. The tilted density is defined as $$\frac{e^{tx} \cdot f(x)}{M(...
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1answer
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Derivation of Distribution Function (CDF) of Gamma Distribution using Poisson Process

I found the following result on Wikipedia relating to the CDF of the Gamma Distribution when the shape parameter is an integer. (Note: there is a slight difference on how I have defined the scale ...
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1answer
76 views

independence of gamma random variables - is this correct?

Suppose $X \sim \Gamma[n_1,\lambda], Y \sim \Gamma[n_2,\lambda]$, and $X+Y \sim \Gamma[n_1 + n_2, \lambda]$ Can we say that $X$ and $Y$ are independent? Here's what I think: Suppose $f,g$ are p.d.f....
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1answer
687 views

How to find alpha and beta for inverse gamma distribution?

I'd like to experiment with using inverse gamma distribution for my data set. If my data was distributed normally, I would have to find sigma and median, and I would be all set. For inverse gamma ...
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1answer
61 views

A proof related to beta and gamma distribution

Please help me to solve the above proof. It is related to beta and gamma function .
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57 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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1answer
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Mean of gamma distribution.

So I was trying to prove the mean result of gamma distribution which is $\frac{\alpha}{\lambda}$. My attempt, $E(X)=\int_{0}^{\infty }x f(x)dx$ $=\int_{0}^{\infty } \frac{\lambda^{\alpha}}{\Gamma (...
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0answers
282 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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1answer
113 views

Is my gamma calculation correct (statistics)

\begin{array}{|c|c|c|c|} \hline \text{Highest Degree}&\text{Don't Believe}& \text{No way to find out}&\text{Some Higher power}&\text{Believe Sometime}&\text{Believe but doubts}&...
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1answer
88 views

Gamma distribution shape

I have a gamma distribution with the following pdf: $$ f(x) = \frac{1}{4} xe^{-0.5x}, x > 0$$ I am trying to determine the shape of the graph without plotting it. I am given a hint ot consider the ...
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54 views

How to find the mean and variance of $Y$ without deriving the distribution of it? [closed]

Let $$Y\mid \Lambda \sim \operatorname{Poisson}(\Lambda)$$ and $$\Lambda \sim \operatorname{Gamma}(\alpha, \beta)$$ where $f_{Y\mid\Lambda}(y\mid\lambda)=\frac{\lambda^y}{y!} e^{-\lambda}, y = 0, 1, 2,...
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(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
348 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=\...
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1answer
626 views

Deriving exponential distribution from sum of two squared normal random variables

Let $X$, $Y$ be i.i.d. random varibales with distribuition $\mathcal{N}(0,1)$ and $Z = X^2 + Y^2$. I'd like to prove based on $X$ and $Y$ pdf's that $Z$ has exponential distribuition.
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1answer
402 views

Chi square and gamma distribution.

I have studied the intuiton of the gamma distribution and have understood the following: Let us suppose that we want to study the probability of waiting time until the $\alpha$-th event occurs. Let $...
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The intuition behind gamma distribution

What is the intuition behind gamma distribution? For instance, I understand how to "construct" Gaussian distribution. This is my intuition: Bernoulli distribution - which is simple concept A ...
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2answers
633 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 ...
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3answers
536 views

Proof that $\sum_{i=1}^nX_i \sim \operatorname{Gamma}(n)$ [duplicate]

How to prove that $\sum_{i=1}^n X_i$ has a $\operatorname{Gamma}(n)$ distribution, where $X_1,\ldots,X_n$ are independent standard exponentials?
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103 views

Does the square of a non central normal distribution follow a gamma distribution?

I know that, as long as the mean of the normal distribution is 0, it can be transformed to a gamma, however, I am not sure about the non-central ones. Maybe it is some kind of non-central chi-...
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1answer
533 views

Solve the system of equations in the maximum likelihood estimation of Gamma distribution parameters

I'm trying to calculate the two parameters of the Gamma distribution by solving the system of two equations obtained by differentiating the ...
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2answers
148 views

approximation for $\Gamma (\alpha) / \Gamma (\beta) $ where $\alpha$ and $\beta$ are arbitrary numbers in $R^{+}$

I am working on implementation of a machine learning method that in part of the algorithm I need to calculate the value of $\Gamma (\alpha) / \Gamma (\beta) $. $\alpha$ and $\beta$ are quite large ...
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1answer
93 views

Is the answer to this Poisson process question correct?

A factory has two production lines which work independently and deliver products to a central packing area. Products arrive from each of the lines at a rate of 1 every two minutes according to a ...
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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 ...
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1answer
1k views

gamma distribution to chi squared transform

I thought If X~gamma($\alpha$, $\beta$) then $\frac{2X}{\alpha}$ ~ $\chi^2_n$ where n=2$\beta$. but when I solve exercise in Mathematical statistics with application (ex9.85e) If Y~Gamma$(\alpha$, $\...
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1answer
52 views

Finding the pdf of this transformed variable

Let $X ∼ Gamma(α, β)$, so that $X$ has pdf $$f(x)=\frac{β^α}{Γ(α)}x^{α-1}exp(−βx)$$ for $x > 0$ and $0$ otherwise Find the pdf of the transformed variable: $Y = X^4$ Ive examined this question ...
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2answers
1k views

Gamma Distribution Sum

I'm trying to show that the sum of three independent gamma distributions $X_1\sim \Gamma(2,5)$, $X_2\sim \Gamma(3,5)$, and $X_3\sim\Gamma(1.5,2.5)$ are a gamma distribution. $Y=X_1+X_2+2X_3$. I've ...
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0answers
76 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 ...