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 $\Lambda_0(t,t_0)=0, \Lambda_1(t,t_0)=1$ | + | 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 |
$$\Lambda_{k+1}(t,t_0) = -\displaystyle\int_{t_0}^t (\ominus \lambda)(\tau) \Lambda_k(\sigma(\tau),t_0) \Delta \tau.$$ | $$\Lambda_{k+1}(t,t_0) = -\displaystyle\int_{t_0}^t (\ominus \lambda)(\tau) \Lambda_k(\sigma(\tau),t_0) \Delta \tau.$$ | ||
| + | |||
| + | 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 footer}} | {{:Probability distributions footer}} | ||
Revision as of 22:13, 14 April 2015
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
Theorem: Let $X$ have the gamma distribution on $\mathbb{T}$. Then $$\mathrm{E}_{\mathbb{T}}(X)=\dfrac{k}{\lambda}.$$
Proof: █
Theorem: Let $X$ have the gamma distribution on $\mathbb{T}$. Then $$\mathrm{Var}_{\mathbb{T}}(X)=\dfrac{k}{\lambda^2}.$$
Proof: █
Probability distributions | ||
| Uniform distribution | Exponential distribution | Gamma distribution |
