Questions tagged [gamma-distribution]

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

A random variable $X$ that is gamma-distributed with shape $k$ and scale $\theta$ is denoted by $$X \sim \Gamma(k, \theta) \equiv \textrm{Gamma}(k, \theta)$$

The probability density function using the shape-scale parametrization is $$f(x;k,\theta) = \frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^k\Gamma(k)} \quad \text{ for } x > 0 \text{ and } k, \theta > 0$$

Here $\Gamma(k)$ is the gamma function evaluated at $k$.

The cumulative distribution function is the regularized gamma function: $$F(x;k,\theta) = \int_0^x f(u;k,\theta)\,du = \frac{\gamma\left(k, \frac{x}{\theta}\right)}{\Gamma(k)}$$

where $\gamma(k, x/ \theta)$ is the lower incomplete gamma function.

518 questions
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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 sequence of Bernoulli trials is a Binomial distribution.…
Andreo
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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.
Amanda R.
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Show that the gamma density integrates to 1.

$$\int_{-\infty}^{\infty} g(t)dt = \int_{0}^{\infty} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} t^{\alpha - 1} e^{-\lambda t} dt = \frac{\lambda}{\Gamma(\alpha)} \int_{0}^{\infty} (\lambda t)^{\alpha - 1} e^{-\lambda t} dt$$ Set $u = \lambda t$: $$=…
Y. Sargis
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Probability density function of a Gamma random variable plus a constant

Suppose $X$ has a Gamma distribution $\Gamma(a,b)$. What is the distribution of $X+c$? where $c$ is a positive constant. EDIT: I know how extract the expression of the pdf and cdf. What I wanted to know is if the resulting distribution is a know…
Nobody
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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 .
Archana kumari
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Loss distribution

please help me solve the following.. Using the gamma distribution with E[X] fixed, show that Var[X] decreases as the shape parameter α increase. Given E[X] =1000 and Var[X] =〖500〗^2, find the appropriate parameter values for the Gamma and…
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Gamma random variable multiplied with a negative constant

If X has a gamma distribution with shape parameter a and rate parameter b, then kX with also be gamma-distributed with rate parameter $\frac{b}{k}$. But this is only true if k is a positive constant. What if k is a negative constant? Will it then…
Thomas
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What is the distribution for a random variable compose with another random variable?

I have a a random variable $X$ with distribution $Po(M)$ where $M$ is a random variable with distribution $Exp(a)$. I want to know what is the "real" distribution of $X$. I was calculating by \begin{equation} P(X=k)=\int_0^{\infty}P(X=k|M=k)\cdot…
Don P.
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Gamma Distribution formulas

Why I am finding two formulas for the gamma distribution The one in my textbook is {1/(beta)^alpha*gamma(alpha)} * x^alpha-1 * exp(-x/beta) The one I found on the internet is {beta^alpha/gamma(alpha)}* x^alpha-1 * exp(-beta*x) (I have plotted…
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Computing the distribution of the sum of independent gamma random variables

I am trying to implement the analytic expression for the distribution of the sum of independent gamma random variables using the expression given in Moschopoulos (1985). More specifically, I would like to implement the analytic expression for the…
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How to calculate the limit $\lim_{n\rightarrow \infty} \sqrt 2 \frac {\Gamma\left(\frac {n+1} 2\right)} {\Gamma\left(\frac {n} 2\right)}-\sqrt n$

$$\lim_{n\rightarrow \infty} \sqrt 2 \frac {\Gamma\left(\frac {n+1} 2\right)} {\Gamma\left(\frac {n} 2\right)}-\sqrt n$$ It seems that using Stirling approximation doesn't work.
zbh2047
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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 find the coefficient A ? My first thought was to…
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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? I have regular customers that come at the…
ASuit
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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?
Joseph
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