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# sympy.stats.StudentT

All Samples(2)  |  Call(1)  |  Derive(0)  |  Import(1)
Create a continuous random variable with a student's t distribution.

The density of the student's t distribution is given by

.. math::
f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)}
{\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)}
\left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}

Parameters(more...)


        def StudentT(name, nu):
r"""
Create a continuous random variable with a student's t distribution.

The density of the student's t distribution is given by

.. math::
f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)}
{\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)}
\left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}

Parameters
==========

nu : Real number, \nu > 0, the degrees of freedom

Returns
=======

A RandomSymbol.

Examples
========

>>> from sympy.stats import StudentT, density, E, variance
>>> from sympy import Symbol, simplify, pprint

>>> nu = Symbol("nu", positive=True)
>>> z = Symbol("z")

>>> X = StudentT("x", nu)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
nu   1
- -- - -
2    2
/     2\
|    z |              /nu   1\
|1 + --|        *gamma|-- + -|
\    nu/              \2    2/
------------------------------
____   ____      /nu\
\/ pi *\/ nu *gamma|--|
\2 /

References
==========

.. [1] http://en.wikipedia.org/wiki/Student_t-distribution
.. [2] http://mathworld.wolfram.com/Studentst-Distribution.html
"""

return rv(name, StudentTDistribution, (nu, ))


from sympy.stats import (P, E, where, density, variance, covariance, skewness,
given, pspace, cdf, ContinuousRV, sample,
Arcsin, Benini, Beta, BetaPrime, Cauchy,
Chi, ChiSquared,
ChiNoncentral, Dagum, Erlang, Exponential,

def test_studentt():
nu = Symbol("nu", positive=True)

X = StudentT('x', nu)
assert density(X)(x) == ((x**2/nu + 1)**(-nu/2 - S.Half)