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 ·