Table 2 PDF and CDF of the functions used in the study.
From: Hydrological projections in the upper reaches of the Yangtze River Basin from 2020 to 2050
Function | CDF | |
|---|---|---|
Gamma | \(f_{G} (x) = \frac{{\alpha^{ - \beta } }}{\Gamma (\beta )}x^{\beta - 1} \exp ( - \frac{x}{\alpha })\) | \(F_{G} (x) = \int_{0}^{x} {\frac{{\alpha^{ - \beta } }}{\Gamma (\beta )}x^{\beta - 1} \exp ( - \frac{x}{\alpha })dx}\) |
Exponential | \(f_{E} (x) = \frac{1}{\alpha }\exp ( - \frac{x}{\alpha })\) | \(F_{E} (x) = 1{\text{ - exp}}( - \frac{x}{\alpha })\) |
GEV | \(f_{V} (x) = \frac{1}{\alpha }(1 + \beta ( - \frac{x - \tau }{\alpha })^{ - 1/\beta - 1} )\exp ( - (1 + \beta (\frac{x - \tau }{\alpha })^{ - 1/\beta } ))\) | \(F_{V} (x) = \exp ( - (1 + \beta (\frac{x - \tau }{\alpha })^{ - 1/\beta } ))\) |
Gumbel | \(f_{U} (x) = \frac{1}{\alpha }\exp ( - \frac{x - \tau }{\alpha })\exp ( - \exp ( - \frac{x - \tau }{\alpha }))\) | \(F_{U} (x) = 1 - \exp ( - \exp ( - \frac{x - \tau }{\alpha }))\) |
GP | \(f_{P} (x) = \frac{1}{\alpha }(1 + \beta (\frac{x - \tau }{\alpha })^{ - 1/\beta - 1} )\) | \(F_{P} (x) = 1 - (1 + \beta (\frac{x - \tau }{\alpha })^{ - 1/\beta } )\) |
