52.2.18 Gumbel Random Variable

Function: pdf_gumbel (x,a,b) ¶

Returns the value at x of the density function of a $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The pdf is

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

Categories: Package distrib ·

Function: cdf_gumbel (x,a,b) ¶

Returns the value at x of the distribution function of a $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The cdf is

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

Categories: Package distrib ·

Function: quantile_gumbel (q,a,b) ¶

Returns the q-quantile of a $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ 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 $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ 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 $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The variance is

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

Categories: Package distrib ·

Function: std_gumbel (a,b) ¶

Returns the standard deviation of a $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The standard deviation is

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

Categories: Package distrib ·

Function: skewness_gumbel (a,b) ¶

Returns the skewness coefficient of a $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ random variable, with $b>0$.

The skewness coefficient is

$$SK[X]=\frac{12\sqrt{6}}{{\pi }^3}\zeta (3)$$

(%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 $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

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

Categories: Package distrib · Package distrib ·

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

Returns a $\mathit{G}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{l}(a,b)$ 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 ·