Table 3 Distribution function and its parameters.
Name of distribution | Function | Parameter |
|---|---|---|
Normal | \(f(x) = \frac{1}{{\sigma \sqrt {2\pi } }}\mathop {\text{e}}\nolimits^{{\left[ {\frac{ - 1}{2}\left( {\frac{x - \mu }{\sigma }} \right)^{2} } \right]}}\) | μ σ |
Log-normal | \(f(x) = \frac{1}{{x\sigma \sqrt {2\pi } }}\mathop e\nolimits^{{\left[ { - \frac{1}{{2\sigma^{2} }}\left( {\log x - \mu } \right)^{2} } \right]}}\) | μ σ |
Logistic | \(f(x) = \frac{1}{\alpha }e^{{\left( {\frac{x - \xi }{\alpha }} \right)\left[ {1 + e^{{\left( {\frac{x - \xi }{\alpha }} \right)}} } \right]^{ - 2} }}\) | ξ α |
Weibull | \(f(x) = \frac{k}{\alpha }\left( {\frac{x}{\alpha }} \right)^{k - 1} \mathop e\nolimits^{{\left[ { - \left( {\frac{x}{\alpha }} \right)^{k} } \right]}}\) | α k |
Gamma | \(f(x) = \frac{{x^{k - 1} }}{{\alpha^{k} \Gamma (k)}}e^{{\left[ { - \frac{x}{\alpha }} \right]}}\) | α k |
EV | \(f(x) = \frac{1}{\sigma }e^{{\left[ { - \frac{{\left( {x - \mu } \right)}}{\sigma } - e^{{\left( { - \frac{(x - \mu )}{\sigma }} \right)}} } \right]}}\) | μ σ |
GEV | \(f(x) = \frac{1}{\alpha }e^{{[ - (1 + kz)^{{ - \frac{1}{k}}} ]}} (1 + kz)^{{ - 1 - \frac{1}{k}}}\) | \(\xi \alpha {\text{ k}}\left( {z = \frac{x - \xi }{\alpha }} \right)\) |
