Difference between revisions of "Gamma distribution"

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Let $\mathbb{T}$ be a time scale. Let $\lambda \in \mathbb{R}$ with $\lambda > 0$ and define $\Lambda_0(t,t_0)=0, \Lambda_1(t,t_0)=1$. The gamma distribution is the [[probability density function]] defined recursively for $k \geq 2$ by the formula
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Let $\mathbb{T}$ be a time scale. Let $\lambda \in \mathbb{R}$ with $\lambda > 0$ and define $\Lambda_0(t,t_0)=0, \Lambda_1(t,t_0)=1$ and
 
$$\Lambda_{k+1}(t,t_0) = -\displaystyle\int_{t_0}^t (\ominus \lambda)(\tau) \Lambda_k(\sigma(\tau),t_0) \Delta \tau.$$
 
$$\Lambda_{k+1}(t,t_0) = -\displaystyle\int_{t_0}^t (\ominus \lambda)(\tau) \Lambda_k(\sigma(\tau),t_0) \Delta \tau.$$
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The gamma [[probability density function]] is defined to be
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$$f_k(t)= \left\{ \begin{array}{ll}
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\dfrac{\lambda}{e_{\lambda}(\sigma(t),0)} \Lambda_k(\sigma(t),0) &; t \geq 0 \\
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0 &; t < 0
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\end{array} \right.$$
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=Properties=
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[[Expected value of gamma distribution]]<br />
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[[Variance of gamma distribution]]
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{{:Probability distributions footer}}
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[[Category:Definition]]

Latest revision as of 14:11, 28 January 2023

Let $\mathbb{T}$ be a time scale. Let $\lambda \in \mathbb{R}$ with $\lambda > 0$ and define $\Lambda_0(t,t_0)=0, \Lambda_1(t,t_0)=1$ and $$\Lambda_{k+1}(t,t_0) = -\displaystyle\int_{t_0}^t (\ominus \lambda)(\tau) \Lambda_k(\sigma(\tau),t_0) \Delta \tau.$$

The gamma probability density function is defined to be $$f_k(t)= \left\{ \begin{array}{ll} \dfrac{\lambda}{e_{\lambda}(\sigma(t),0)} \Lambda_k(\sigma(t),0) &; t \geq 0 \\ 0 &; t < 0 \end{array} \right.$$

Properties

Expected value of gamma distribution
Variance of gamma distribution

Probability distributions

Uniform distributionExponential distributionGamma distribution