Updating sympy.stats.crv_types for extreme value distributions

sympy/core/tests/test_args.py

@@ -1359,7 +1359,7 @@ def test_sympy__stats__crv_types__GammaDistribution():
 
 def test_sympy__stats__crv_types__GumbelDistribution():
     from sympy.stats.crv_types import GumbelDistribution
-    assert _test_args(GumbelDistribution(1, 1))
+    assert _test_args(GumbelDistribution(1, 1, False))
 
 def test_sympy__stats__crv_types__GompertzDistribution():
     from sympy.stats.crv_types import GompertzDistribution

sympy/stats/crv_types.py

@@ -50,7 +50,7 @@
 from sympy import (log, sqrt, pi, S, Dummy, Interval, sympify, gamma,
                    Piecewise, And, Eq, binomial, factorial, Sum, floor, Abs,
                    Lambda, Basic, lowergamma, erf, erfi,  erfinv, I, hyper,
-                   uppergamma, sinh, atan, Ne, expint)
+                   uppergamma, sinh, atan, Ne, expint, Integral)
 
 from sympy import beta as beta_fn
 from sympy import cos, sin, tan, atan, exp, besseli, besselj, besselk
@@ -1563,58 +1563,78 @@ def GammaInverse(name, a, b):
     return rv(name, GammaInverseDistribution, (a, b))
 
 #-------------------------------------------------------------------------------
-# Gumbel distribution --------------------------------------------------------
+# Gumbel distribution (Maximum and Minimum) --------------------------------------------------------
 
 
 class GumbelDistribution(SingleContinuousDistribution):
-    _argnames = ('beta', 'mu')
+    _argnames = ('beta', 'mu', 'minimum')
 
     set = Interval(-oo, oo)
 
     @staticmethod
-    def check(beta, mu):
+    def check(beta, mu, minimum):
         _value_check(beta > 0, "Scale parameter beta must be positive.")
 
     def pdf(self, x):
         beta, mu = self.beta, self.mu
         z = (x - mu)/beta
-        return (1/beta)*exp(-(z + exp(-z)))
+        f_max = (1/beta)*exp(-z - exp(-z))
+        f_min = (1/beta)*exp(z - exp(z))
+        return Piecewise((f_min, self.minimum), (f_max, not self.minimum))
 
     def _cdf(self, x):
         beta, mu = self.beta, self.mu
-        return exp(-exp((mu - x)/beta))
+        z = (x - mu)/beta
+        F_max = exp(-exp(-z))
+        F_min = 1 - exp(-exp(z))
+        return Piecewise((F_min, self.minimum), (F_max, not self.minimum))
 
     def _characteristic_function(self, t):
-        return gamma(1 - I*self.beta*t) * exp(I*self.mu*t)
+        cf_max = gamma(1 - I*self.beta*t) * exp(I*self.mu*t)
+        cf_min = gamma(1 + I*self.beta*t) * exp(I*self.mu*t)
+        return Piecewise((cf_min, self.minimum), (cf_max, not self.minimum))
 
     def _moment_generating_function(self, t):
-        return gamma(1 - self.beta*t) * exp(self.mu*t)
+        mgf_max = gamma(1 - self.beta*t) * exp(self.mu*t)
+        mgf_min = gamma(1 + self.beta*t) * exp(self.mu*t)
+        return Piecewise((mgf_min, self.minimum), (mgf_max, not self.minimum))
 
-def Gumbel(name, beta, mu):
+def Gumbel(name, beta, mu, minimum=False):
     r"""
     Create a Continuous Random Variable with Gumbel distribution.
 
     The density of the Gumbel distribution is given by
 
+    For Maximum
+
     .. math::
         f(x) := \dfrac{1}{\beta} \exp \left( -\dfrac{x-\mu}{\beta}
                 - \exp \left( -\dfrac{x - \mu}{\beta} \right) \right)
 
     with :math:`x \in [ - \infty, \infty ]`.
 
+    For Minimum
+
+    .. math::
+        f(x) := \frac{e^{- e^{\frac{- \mu + x}{\beta}} + \frac{- \mu + x}{\beta}}}{\beta}
+
+    with :math:`x \in [ - \infty, \infty ]`.
+
     Parameters
     ==========
 
-    mu: Real number, 'mu' is a location
-    beta: Real number, 'beta > 0' is a scale
+    mu : Real number, 'mu' is a location
+    beta : Real number, 'beta > 0' is a scale
+    minimum : Boolean, by default, False, set to True for enabling minimum distribution
 
     Returns
-    ==========
+    =======
 
     A RandomSymbol
 
     Examples
-    ==========
+    ========
+
     >>> from sympy.stats import Gumbel, density, E, variance, cdf
     >>> from sympy import Symbol, simplify, pprint
     >>> x = Symbol("x")
@@ -1624,16 +1644,18 @@ def Gumbel(name, beta, mu):
     >>> density(X)(x)
     exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta
     >>> cdf(X)(x)
-    exp(-exp((mu - x)/beta))
+    exp(-exp(-(-mu + x)/beta))
 
     References
     ==========
 
     .. [1] http://mathworld.wolfram.com/GumbelDistribution.html
     .. [2] https://en.wikipedia.org/wiki/Gumbel_distribution
+    .. [3] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_max.html
+    .. [4] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_min.html
 
     """
-    return rv(name, GumbelDistribution, (beta, mu))
+    return rv(name, GumbelDistribution, (beta, mu, minimum))
 
 #-------------------------------------------------------------------------------
 # Gompertz distribution --------------------------------------------------------

sympy/stats/tests/test_continuous_rv.py

@@ -631,9 +631,15 @@ def test_gumbel():
     beta = Symbol("beta", positive=True)
     mu = Symbol("mu")
     x = Symbol("x")
+    y = Symbol("y")
     X = Gumbel("x", beta, mu)
-    assert str(density(X)(x)) == 'exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta'
-    assert cdf(X)(x) == exp(-exp((mu - x)/beta))
+    Y = Gumbel("y", beta, mu, minimum=True)
+    assert density(X)(x).expand() == \
+    exp(mu/beta)*exp(-x/beta)*exp(-exp(mu/beta)*exp(-x/beta))/beta
+    assert density(Y)(y).expand() == \
+    exp(-mu/beta)*exp(y/beta)*exp(-exp(-mu/beta)*exp(y/beta))/beta
+    assert cdf(X)(x).expand() == \
+    exp(-exp(mu/beta)*exp(-x/beta))
 
 
 def test_kumaraswamy():