#### 52.2.18 Gumbel Random Variable

Function: pdf_gumbel (x,a,b)

  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 $$ f(x; a, b) = {1\over b} \exp\left[{a-x\over b} - \exp\left({a-x\over b}\right)\right] $$

  Categories:Package distrib ·

Function: cdf_gumbel (x,a,b)

  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 $$ F(x; a, b) = \exp\left[-\exp\left({a-x\over b}\right)\right] $$

  Categories:Package distrib ·

Function: quantile_gumbel (q,a,b)

  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")`.

  Categories:Package distrib ·

Function: mean_gumbel (a,b)

  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`.

  Categories:Package distrib ·

Function: var_gumbel (a,b)

  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 $$ V[X] = {\pi^2\over 6} b^2 $$

  Categories:Package distrib ·

Function: std_gumbel (a,b)

  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 $$ D[X] = {\pi \over \sqrt{6}} b $$

  Categories:Package distrib ·

Function: skewness_gumbel (a,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.

  Categories:Package distrib ·

Function: kurtosis_gumbel (a,b)

  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 $$ KU[X] = {12\over 5} $$

  Categories:Package distrib ·Package distrib ·

Function: random_gumbel (a,b)
`random_gumbel` (a,b,n)

  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")`.

  Categories:Package distrib ·Random numbers ·