1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions


Probability Density Function 
The extreme value type I distribution has two forms. One is based on
the smallest extreme and the other is based on the largest extreme. We
call these the minimum and maximum cases, respectively. Formulas and
plots for both cases are given. The extreme value type I distribution is
also referred to as the Gumbel distribution.
The general formula for the probability density function of the Gumbel (minimum) distribution is \( f(x) = \frac{1} {\beta} e^{\frac{x\mu}{\beta}}e^{e^{\frac{x\mu} {\beta}}} \) where μ is the location parameter and β is the scale parameter. The case where μ = 0 and β = 1 is called the standard Gumbel distribution. The equation for the standard Gumbel distribution (minimum) reduces to \( f(x) = e^{x}e^{e^{x}} \) The following is the plot of the Gumbel probability density function for the minimum case.
The general formula for the probability density function of the Gumbel (maximum) distribution is \( f(x) = \frac{1}{\beta} e^{\frac{x\mu}{\beta}}e^{e^{\frac{x\mu} {\beta}}} \) where μ is the location parameter and β is the scale parameter. The case where μ = 0 and β = 1 is called the standard Gumbel distribution. The equation for the standard Gumbel distribution (maximum) reduces to \( f(x) = e^{x}e^{e^{x}} \) The following is the plot of the Gumbel probability density function for the maximum case. \( f(x) = e^{x}e^{e^{x}} \) Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. 

Cumulative Distribution Function 
The formula for the cumulative distribution
function of the Gumbel distribution (minimum) is
\( F(x) = 1  e^{e^{x}} \) The following is the plot of the Gumbel cumulative distribution function for the minimum case.
The formula for the cumulative distribution function of the Gumbel distribution (maximum) is \( F(x) = e^{e^{x}} \) The following is the plot of the Gumbel cumulative distribution function for the maximum case.


Percent Point Function 
The formula for the percent point
function of the Gumbel distribution (minimum) is
\( G(p) = \ln(\ln(\frac{1} {1  p})) \) The following is the plot of the Gumbel percent point function for the minimum case.
The formula for the percent point function of the Gumbel distribution (maximum) is \( G(p) = \ln(\ln(\frac{1} {p})) \) The following is the plot of the Gumbel percent point function for the maximum case.


Hazard Function 
The formula for the hazard
function of the Gumbel distribution (minimum) is
\( h(x) = e^{x} \) The following is the plot of the Gumbel hazard function for the minimum case.
The formula for the hazard function of the Gumbel distribution (maximum) is \( h(x) = \frac{e^{x}} {e^{e^{x}}  1} \) The following is the plot of the Gumbel hazard function for the maximum case.


Cumulative Hazard Function 
The formula for the cumulative hazard
function of the Gumbel distribution (minimum) is
\( H(x) = e^{x} \) The following is the plot of the Gumbel cumulative hazard function for the minimum case.
The formula for the cumulative hazard function of the Gumbel distribution (maximum) is \( H(x) = \ln(1  e^{e^{x}}) \) The following is the plot of the Gumbel cumulative hazard function for the maximum case.


Survival Function 
The formula for the survival
function of the Gumbel distribution (minimum) is
\( S(x) = e^{e^{x}} \) The following is the plot of the Gumbel survival function for the minimum case.
The formula for the survival function of the Gumbel distribution (maximum) is \( S(x) = 1  e^{e^{x}} \) The following is the plot of the Gumbel survival function for the maximum case.


Inverse Survival Function 
The formula for the inverse
survival function of the Gumbel distribution (minimum) is
\( Z(p) = \ln(\ln(\frac{1} {p})) \) The following is the plot of the Gumbel inverse survival function for the minimum case.
The formula for the inverse survival function of the Gumbel distribution (maximum) is \( Z(p) = \ln(\ln(\frac{1} {1p})) \) The following is the plot of the Gumbel inverse survival function for the maximum case.


Common Statistics 
The formulas below are for the maximum order statistic case.


Parameter Estimation 
The method of moments estimators of the Gumbel (maximum) distribution
are
\( \tilde{\beta} = \frac{s\sqrt{6}} {\pi} \) \( \tilde{\mu} = \bar{X}  0.5772 \tilde{\beta} = \bar{X}  0.45006 s \) where \( \bar{X} \) and s are the sample mean and standard deviation, respectively. The method of moments estimators of the Gumbel (minimum) distribution are \( \tilde{\beta} = \frac{s\sqrt{6}} {\pi} \) \( \tilde{\mu} = \bar{X} + 0.5772 \tilde{\beta} = \bar{X} + 0.45006 s \) where \( \bar{X} \) and s are the sample mean and standard deviation, respectively. The maximum likelihood estimates for the maximum case are the solution to the following simultaneous equations \( \bar{x}  \frac{\sum_{i=1}^{n}{x_i \exp(x_i/\hat{\beta})}} {\sum_{i=1}^{n}{\exp(x_i/\hat{\beta})}}  \hat{\beta} = 0 \) \( \hat{\beta} \log \left( \frac{1}{n} \sum_{i=1}^{n}{\exp(x_i/\hat{\beta})} \right)  \hat{\mu} = 0 \) For the minimum case, replace \(x_i\) with \(x_i\) in the above equations. These equations need to be solved numerically and this is typically accomplished by using statistical software packages. 

Software  Some general purpose statistical software programs support at least some of the probability functions for the extreme value type I distribution. 