# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...