2.7. Annex A Probability distributions

In the following, the parametrization of the probability distributions introduced in Section 2.4.4 (Model development: Statistical methods) and used in Section 2.5 (Bayesian methods) of this part, is given as a common reference.

Exponential distribution

\(f_{X}(x) = \ \frac{\beta}{\alpha} \cdot \left( \frac{x}{\alpha} \right)^{\beta - 1}\mathrm{\exp}\left\lbrack - \left( \frac{x}{\alpha} \right)^{\beta} \right\rbrack\)	\(X \geq 0\)	\(E\left\lbrack X \right\rbrack = \alpha \cdot \Gamma\left( 1 + \frac{1}{\beta} \right)\)
\(F_{X}\left( x \right) = \ 1 - \mathrm{\exp}\left\lbrack - \left( \frac{x}{\alpha} \right)^{\beta} \right\rbrack\)	\(\alpha > 0\)	\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \alpha \cdot \sqrt{\Gamma\left( 1 + \frac{2}{\beta} \right) - \Gamma^{2}\left( 1 + \frac{1}{\beta} \right)}\)

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 \(\pm 5\%\).

Gamma distribution

\(f_{X}(x) = \ \frac{\beta^{\alpha}}{\Gamma\left( \alpha \right)} \cdot x^{\alpha - 1} \cdot \exp( - \beta x)\)	\(X \geq 0\)	\(E\left\lbrack X \right\rbrack = \frac{\alpha}{\beta}\)
\(F_{X}\left( x \right) = \frac{\Gamma\left( \alpha,\beta x \right)}{\Gamma\left( \alpha \right)}\)	\(\alpha > 0 , \beta > 0\)	\(\alpha > 0\)\( | \)\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{\sqrt{\alpha}}{\beta}\)

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

\(f_{X}(x) = \ \frac{1}{\sigma\sqrt{2\pi}} \cdot \exp\left\lbrack - \frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^{2} \right\rbrack\)	\(X\mathbb{\in R}\)	\(E\left\lbrack X \right\rbrack = \mu\)
\(F_{X}\left( x \right) = \ \Phi\left( \frac{x - \mu}{\sigma} \right)\)	\(\mu\mathbb{\in R} , \sigma > 0\)	\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \sigma\)

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

\(f_{X}(x) = \ \frac{1}{\sigma\sqrt{2\pi}} \cdot \exp\left\lbrack - \frac{1}{2}\left( \frac{\ln x - \mu}{\sigma} \right)^{2} \right\rbrack\)	\(X \geq 0\)	\(E\left\lbrack X \right\rbrack = \exp\left( \mu + \frac{\sigma^{2}}{2} \right)\)
\(F_{X}\left( x \right) = \ \Phi\left( \frac{\ln x - \mu}{\sigma} \right)\)	\(\mu\mathbb{\in R} , \sigma > 0\)	\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \exp\left( \mu + \frac{\sigma^{2}}{2} \right)\sqrt{\exp\left( \sigma^{2} \right) - 1}\)

Geometric distribution

\(p_{X}\left( x \right) = p \cdot \left( 1 - p \right)^{x - 1}\)	\(X = 1,2,3,\ldots\)	\(E\left\lbrack X \right\rbrack = 1/p\)
\(P_{X}\left( x \right) = 1 - \left( 1 - p \right)^{x}\)	\(0 < p < 1\)	\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{\sqrt{1 - p}}{p}\)

Binomial distribution

\(p_{X}\left( X \right) = \ \begin{pmatrix} n \\ x \\ \end{pmatrix}p^{x}\left( 1 - p \right)^{n - x}\)	\(X = 1,2,\ldots n\)	\(E\left\lbrack X \right\rbrack = np\)
\(P_{X}\left( x \right) = \sum_{i = 0}^{x}{\begin{pmatrix}n \\ i \\\end{pmatrix}p^{i}\left( 1 - p \right)^{n - i}}\)	\(0 < p < 1\)	\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \sqrt{\text{np}\left( 1 - p \right)}\)

Beta distribution

\(f_{X}\left( x \right) = \frac{1}{B\left( \alpha,\beta \right)}x^{\alpha - 1}\left( 1 - x \right)^{\beta - 1}\)	\(0 \leq X \leq 1\)	\(E\left\lbrack X \right\rbrack = \frac{\alpha}{\left( \alpha + \beta \right)}\)
\(f_{X}\left( x \right) = \frac{B\left( x;\alpha,\beta \right)}{B\left( \alpha,\beta \right)}\)	\(\alpha > 0 , \beta > 0\)	\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{1}{\alpha + \beta} \cdot \sqrt{\frac{\text{αβ}}{\alpha + \beta + 1}}\)

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

\(f_{X}(x) = \ \frac{\beta^{\alpha}}{\Gamma\left( \alpha \right)} \cdot \frac{\exp( - \beta x)}{x^{\alpha + 1}}\)	\(X \geq 0\)	\(E\left\lbrack X \right\rbrack = \frac{\beta}{\left( \alpha - 1 \right)}\)
\(F_{X}\left( x \right) = \frac{\Gamma\left( \alpha,\beta/x \right)}{\Gamma\left( \alpha \right)}\)	\(\alpha > 0, \beta > 0\)	\(\sqrt{\mathrm{\text{Var}}\left\lbrack X \right\rbrack} = \frac{\beta}{\left( \left( \alpha - 1 \right) \cdot \sqrt{\alpha - 2} \right)}\)