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Returns the value at x of the density function of a
\({\it Gumbel}(a,b)\) random variable, with b>0. To make use of this function, write first load("distrib").
The pdf is $$ f(x; a, b) = {1\over b} \exp\left[{a-x\over b} - \exp\left({a-x\over b}\right)\right] $$
Returns the value at x of the distribution function of a
\({\it Gumbel}(a,b)\) random variable, with b>0. To make use of this function, write first load("distrib").
The cdf is $$ F(x; a, b) = \exp\left[-\exp\left({a-x\over b}\right)\right] $$
Returns the q-quantile of a
\({\it Gumbel}(a,b)\) random variable, with b>0; in other words, this is the inverse of cdf_gumbel. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").
Returns the mean of a \({\it Gumbel}(a,b)\) random variable, with b>0.
The mean is $$ E[X] = a+b\gamma $$
(%i1) load ("distrib")$
(%i2) mean_gumbel(a,b); (%o2) %gamma b + a
where symbol %gamma stands for the Euler-Mascheroni constant. See also %gamma.
Returns the variance of a
\({\it Gumbel}(a,b)\) random variable, with b>0. To make use of this function, write first load("distrib").
The variance is $$ V[X] = {\pi^2\over 6} b^2 $$
Returns the standard deviation of a
\({\it Gumbel}(a,b)\) random variable, with b>0. To make use of this function, write first load("distrib").
The standard deviation is $$ D[X] = {\pi \over \sqrt{6}} b $$
Returns the skewness coefficient of a \({\it Gumbel}(a,b)\) random variable, with b>0.
The skewness coefficient is $$ SK[X] = {12\sqrt{6}\over \pi^3} \zeta(3) $$
(%i1) load ("distrib")$
(%i2) skewness_gumbel(a,b);
3/2
2 6 zeta(3)
(%o2) --------------
3
%pi
where zeta stands for the Riemann’s zeta function.
Returns the kurtosis coefficient of a
\({\it Gumbel}(a,b)\) random variable, with b>0. To make use of this function, write first load("distrib").
The kurtosis coefficient is $$ KU[X] = {12\over 5} $$
Returns a
\({\it Gumbel}(a,b)\) random variate, with b>0. Calling random_gumbel with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load("distrib").