Questions tagged [gamma-distribution]

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

284 questions
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Is the moment generating function of the gamma density $g(t)=(\frac{\lambda}{\lambda - t})^n$?

My book defines the gamma density as the following: $$f_X(x)=\lambda (\lambda x)^{n-1}e^{-\lambda x}/(n-1)!$$ And has the moment generating function of this density as $\frac{\lambda}{\lambda +t}$. Is ...
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Transforming sum of n exponential distribution to a Poisson distribution

Let $X_1,...,X_n$ be i.i.d exponential random variable with mean $\lambda$ $S=X_1+...+X_n$ So by finding the mgf of S, we get that $S \sim \operatorname{Gamma}(n,\lambda)$ The problem I am stuck ...
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A relationship between Poisson distribution and gamma distribution

We define $N(t)$ to be number of events in the interval $[0,t]$. We assume that $N(t) \sim P(\lambda t)$ for $\lambda > 0$. Let $X$ be the waiting time until the $n$-th event, we need to prove that ...
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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 ...
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Introduction of shape parameters in the formulation of probability distribution

I'm familiar with the definition of location, scale, and shape parameters, and the type of distributions they parametrized. I'm interested in understanding how shape parameters became part of the ...
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A process converging to a certain distribution [closed]

How is it possible to build a discrete-time stochastic process so that converges to a specific distribution at equilibrium, for example an exponential distribution or a gamma distribution ?
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estimate Gamma parameters based on mean and variance

I am following these two approaches (which are the same)this and this, to estimate the two parameters of Gamma dist based on mean and var. I am not sure why I cannot get the same mean and var from ...
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Method of moments with a Gamma distribution

I'm more so confused on a specific step in obtaining the MOM than completely obtaining the MOM: Given a random sample of $Y_1 , Y_2,..., Y_i$ ~ $Gamma (\alpha , \beta)$ find the MOM So I found the ...