(annexII_A)= # Annex A Probability distributions In the following, the parametrization of the probability distributions introduced in {numref}`method_5_4` (Model development: {term}`Statistical methods `) and used in {numref}`method_6` (Bayesian methods) of this part, is given as a common reference. **Exponential distribution** ```{list-table} * - $f_{X}(x) = \ \frac{\beta}{\alpha} \cdot \left( \frac{x}{\alpha} \right)^{\beta - 1}\mathrm{\exp}\left\lbrack - \left( \frac{x}{\alpha} \right)^{\beta} \right\rbrack$ - $X \geq 0$ - $E\left\lbrack X \right\rbrack = \alpha \cdot \Gamma\left( 1 + \frac{1}{\beta} \right)$ * - $F_{X}\left( x \right) = \ 1 - \mathrm{\exp}\left\lbrack - \left( \frac{x}{\alpha} \right)^{\beta} \right\rbrack$ - $\alpha > 0$ - $\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \alpha \cdot \sqrt{\Gamma\left( 1 + \frac{2}{\beta} \right) - \Gamma^{2}\left( 1 + \frac{1}{\beta} \right)}$ ``` The mean approximation bias of $\beta \approx \left( v_{x} \right)^{- 1.09}$ is shown in the diagram to the right. For coefficients of variation $0.05 \leq v_{X} \leq 1.35$, the approximation error is smaller then $\pm 5\%$. ```{figure} pictures/annexII_A_1.png --- width: 600px ``` **Gamma distribution** ```{list-table} * - $f_{X}(x) = \ \frac{\beta^{\alpha}}{\Gamma\left( \alpha \right)} \cdot x^{\alpha - 1} \cdot \exp( - \beta x)$ - $X \geq 0$ - $E\left\lbrack X \right\rbrack = \frac{\alpha}{\beta}$ * - $F_{X}\left( x \right) = \frac{\Gamma\left( \alpha,\beta x \right)}{\Gamma\left( \alpha \right)}$ - $\alpha > 0 , \beta > 0$ - $\alpha > 0$$ | $$\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{\sqrt{\alpha}}{\beta}$ ``` Gamma function: $\Gamma\left( \alpha \right) = \int_{0}^{\infty}{t^{\alpha - 1}\mathrm{\exp}\left( - t \right)\mathrm{d}t}$ Lower incomplete gamma function: $\Gamma\left( \alpha,\beta x \right) = \int_{0}^{\text{βx}}{t^{\alpha - 1}\mathrm{\exp}\left( - t \right)\mathrm{d}t}$ **Normal distribution** ```{list-table} * - $f_{X}(x) = \ \frac{1}{\sigma\sqrt{2\pi}} \cdot \exp\left\lbrack - \frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^{2} \right\rbrack$ - $X\mathbb{\in R}$ - $E\left\lbrack X \right\rbrack = \mu$ * - $F_{X}\left( x \right) = \ \Phi\left( \frac{x - \mu}{\sigma} \right)$ - $\mu\mathbb{\in R} , \sigma > 0$ - $\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \sigma$ ``` Standard normal distribution: $\Phi\left( x \right) = \ \frac{1}{\sqrt{2\pi}} \cdot \int_{- \infty}^{x}{\exp\left\lbrack - \frac{t^{2}}{2} \right\rbrack}\mathrm{d}t$ **Lognormal distribution** ```{list-table} * - $f_{X}(x) = \ \frac{1}{\sigma\sqrt{2\pi}} \cdot \exp\left\lbrack - \frac{1}{2}\left( \frac{\ln x - \mu}{\sigma} \right)^{2} \right\rbrack$ - $X \geq 0$ - $E\left\lbrack X \right\rbrack = \exp\left( \mu + \frac{\sigma^{2}}{2} \right)$ * - $F_{X}\left( x \right) = \ \Phi\left( \frac{\ln x - \mu}{\sigma} \right)$ - $\mu\mathbb{\in R} , \sigma > 0$ - $\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \exp\left( \mu + \frac{\sigma^{2}}{2} \right)\sqrt{\exp\left( \sigma^{2} \right) - 1}$ ``` **Geometric distribution** ```{list-table} * - $p_{X}\left( x \right) = p \cdot \left( 1 - p \right)^{x - 1}$ - $X = 1,2,3,\ldots$ - $E\left\lbrack X \right\rbrack = 1/p$ * - $P_{X}\left( x \right) = 1 - \left( 1 - p \right)^{x}$ - $0 < p < 1$ - $\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{\sqrt{1 - p}}{p}$ ``` **Binomial distribution** ```{list-table} * - $p_{X}\left( X \right) = \ \begin{pmatrix} n \\ x \\ \end{pmatrix}p^{x}\left( 1 - p \right)^{n - x}$ - $X = 1,2,\ldots n$ - $E\left\lbrack X \right\rbrack = np$ * - $P_{X}\left( x \right) = \sum_{i = 0}^{x}{\begin{pmatrix}n \\ i \\\end{pmatrix}p^{i}\left( 1 - p \right)^{n - i}}$ - $0 < p < 1$ - $\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \sqrt{\text{np}\left( 1 - p \right)}$ ``` **Beta distribution** ```{list-table} * - $f_{X}\left( x \right) = \frac{1}{B\left( \alpha,\beta \right)}x^{\alpha - 1}\left( 1 - x \right)^{\beta - 1}$ - $0 \leq X \leq 1$ - $E\left\lbrack X \right\rbrack = \frac{\alpha}{\left( \alpha + \beta \right)}$ * - $f_{X}\left( x \right) = \frac{B\left( x;\alpha,\beta \right)}{B\left( \alpha,\beta \right)}$ - $\alpha > 0 , \beta > 0$ - $\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{1}{\alpha + \beta} \cdot \sqrt{\frac{\text{αβ}}{\alpha + \beta + 1}}$ ``` Beta function: $B\left( \alpha,\beta \right) = \Gamma\left( \alpha \right) \cdot \Gamma\left( \beta \right)/\Gamma\left( \alpha + \beta \right)$ Incomplete beta function: $B\left( x;\alpha,\beta \right) = \int_{0}^{x}{t^{\alpha - 1}\left( 1 - t \right)^{\beta - 1}\mathrm{d}t}$ **Inverse gamma distribution** ```{list-table} * - $f_{X}(x) = \ \frac{\beta^{\alpha}}{\Gamma\left( \alpha \right)} \cdot \frac{\exp( - \beta x)}{x^{\alpha + 1}}$ - $X \geq 0$ - $E\left\lbrack X \right\rbrack = \frac{\beta}{\left( \alpha - 1 \right)}$ * - $F_{X}\left( x \right) = \frac{\Gamma\left( \alpha,\beta/x \right)}{\Gamma\left( \alpha \right)}$ - $\alpha > 0, \beta > 0$ - $\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{\beta}{\left( \left( \alpha - 1 \right) \cdot \sqrt{\alpha - 2} \right)}$ ```