Table 2 Description of various probability distribution functions.
From: A systematic approach to modeling monthly maximum temperature and total rainfall in Kenya
Distributions | Probability density functions | Ranges | Parameters |
|---|---|---|---|
Normal | \(f(x) = \frac{1}{\sigma \sqrt{2\pi }} e^{-\frac{1}{2}\left( \frac{x-\mu }{\sigma }\right) ^2}\) | \(-\infty< x < \infty\) | \(\sigma\): standard deviation |
\(\mu\): mean | |||
Lognormal | \(f(x,\mu ,\sigma ) = \frac{1}{(x) \sigma \sqrt{2\pi }} \exp \left[ {-\frac{1}{2}\left( \frac{\ln (x) - \mu }{\sigma }\right) ^2} \right]\) | \(x > 0\) | \(\mu\): scale parameter |
\(\sigma\): shape parameter | |||
Weibull | \(f(x,\alpha ,\beta ) = \frac{\alpha }{\beta } \left( \frac{x}{\beta } \right) ^{\alpha -1} \exp \left( -\left( \frac{x}{\beta } \right) ^\alpha \right)\) | \(x > 0\) | \(\alpha , \beta > 0\) |
\(\alpha\): shape parameter | |||
\(\beta\): scale parameter | |||
GEV | \(f(x;\mu , \sigma , \xi ) = \frac{1}{\sigma } \left[ 1 + \xi \left( \frac{x - \mu }{\sigma } \right) \right] ^{-\frac{1}{\xi }-1} \exp \left( - \left[ 1 + \xi \left( \frac{x - \mu }{\sigma } \right) \right] ^{-\frac{1}{\xi }} \right)\) | \(x \in \left[ \mu -\frac{\sigma }{\xi }, + \infty \right] \text {if} > 0\) | \(\mu\): location parameter |
\(x \in \left[ - \infty , + \infty \right] \text { if} = 0\) | \(\xi\): shape parameter | ||
\(x \in \left[ -\infty , \mu -\frac{\sigma }{\xi } \right] \text {if} < 0\) | \(\sigma\): scale parameter \(\sigma > 0\) | ||
\(\mu \in \mathbb {R}\) | |||
Exponential | \(f(x,\lambda ) = \frac{1}{\lambda } \exp {\left[ -\frac{x}{\lambda } \right] }\) | \(x \ge 0\) | \(\lambda\): rate parameter |
\(x \ge 0\) | |||
Gamma | \(f(x,\alpha ,\beta ) = \frac{1}{\beta ^\alpha \Gamma (\alpha )} x^{\alpha - 1} \exp {\left( -\frac{x}{\beta }\right) }\) | \(0< x < \infty\) | \(\alpha\): shape parameter |
\(\beta\): scale parameter | |||
Logistic | \(f(x; \mu , s) = \frac{\exp \left( -\frac{x - \mu }{s}\right) }{s \left( 1 + \exp \left( -\frac{x - \mu }{s}\right) \right) ^2}\) | \(-\infty< x < \infty\) | \(\mu\): location parameter |
s: scale parameter | |||
Gumbel | \(f(x; \alpha , \beta ) = \frac{1}{\beta } \exp \left[ -\frac{x - \alpha }{\beta } - \exp \left( -\frac{x - \alpha }{\beta }\right) \right]\) | \(-\infty< x < \infty\) | \(\alpha\): location parameter |
\(\beta\): scale parameter | |||
\(\xi = 0\): shape parameter | |||
Uniform | \(f(x; a, b) = \frac{1}{b - a}\) | \(a \le x \le b\) | a: Lower bound |
b: Upper bound | |||
GPD | \(\frac{1}{\sigma } \left( 1 + \xi \frac{x - \mu }{\sigma }\right) ^{-\frac{1}{\xi } - 1} \text { if } \xi \ne 0\), \(\frac{1}{\sigma } \exp \left( -\frac{x - \mu }{\sigma }\right) \text { if } \xi = 0.\), | \(x \ge \mu \text { if } \xi \ge 0;\) \(\mu \le x \le \mu - \frac{\sigma }{\xi } \text { if } \xi < 0\) \(\mu \le x \le \infty \text { if } \xi = 0\) | \(\mu\): location parameter |
\(\sigma\): scale parameter | |||
\(\xi\): shape parameter |
