Gamma distribution

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