A (right-skewed) Gumbel continuous random variable.

Continuous random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:


x : array-like


q : array-like

lower or upper tail probability

<shape(s)> : array-like

shape parameters

loc : array-like, optional

location parameter (default=0)

scale : array-like, optional

scale parameter (default=1)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : string, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)


gumbel_r.rvs(loc=0,scale=1,size=1) :

  • random variates

gumbel_r.pdf(x,loc=0,scale=1) :

  • probability density function

gumbel_r.cdf(x,loc=0,scale=1) :

  • cumulative density function

gumbel_r.sf(x,loc=0,scale=1) :

  • survival function (1-cdf — sometimes more accurate)

gumbel_r.ppf(q,loc=0,scale=1) :

  • percent point function (inverse of cdf — percentiles)

gumbel_r.isf(q,loc=0,scale=1) :

  • inverse survival function (inverse of sf)

gumbel_r.stats(loc=0,scale=1,moments=’mv’) :

  • mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’)

gumbel_r.entropy(loc=0,scale=1) :

  • (differential) entropy of the RV.,loc=0,scale=1) :

  • Parameter estimates for gumbel_r data

Alternatively, the object may be called (as a function) to fix the shape, :

location, and scale parameters returning a “frozen” continuous RV object: :

rv = gumbel_r(loc=0,scale=1) :

  • frozen RV object with the same methods but holding the given shape, location, and scale fixed


>>> import matplotlib.pyplot as plt
>>> numargs = gumbel_r.numargs
>>> [ <shape(s)> ] = [0.9,]*numargs
>>> rv = gumbel_r(<shape(s)>)

Display frozen pdf

>>> x = np.linspace(0,np.minimum(rv.dist.b,3))
>>> h=plt.plot(x,rv.pdf(x))

Check accuracy of cdf and ppf

>>> prb = gumbel_r.cdf(x,<shape(s)>)
>>> h=plt.semilogy(np.abs(x-gumbel_r.ppf(prb,c))+1e-20)

Random number generation

>>> R = gumbel_r.rvs(size=100)

Right-skewed Gumbel (Log-Weibull, Fisher-Tippett, Gompertz) distribution

gumbel_r.pdf(x) = exp(-(x+exp(-x)))

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