pNonCentralStudent

pNonCentralStudent(nu, delta, x, tail=False)

Cumulative distribution function of a NonCentralStudent distribution.

Parameters:
nufloat, \nu > 0

The \nu parameter.

deltafloat, \delta > 0

The \nu parameter.

xfloat

Location.

tailbool, optional

Tail flag. Default value is False. If True, the complementary CDF is computed.

Returns:
pfloat

The CDF or the complementary CDF at x.

Notes

The position parameter \gamma is equal to zero.

The probability density function is defined as:

f_X(x) = \frac{exp \left(-\delta^2 / 2 \right)} {\sqrt{\nu\pi} \Gamma \left(\frac{\nu}{2} \right)} \left(\frac{\nu}{\nu + x^2}\right) ^ {\frac{\nu + 1}{2}} \sum_{j=0}^{\infty} \frac{\Gamma \left(\frac{\nu + j + 1}{2}\right)}{\Gamma(j + 1)} \left(\delta(x - \gamma) \sqrt{\frac{2}{\nu + x^2}}\right) ^ j, \quad x \in \Rset

With \Gamma denotes Euler’s Gamma function Gamma().

We use Viktor Witkovsky’s algorithm described in [witkovsky2003].

Examples

>>> import openturns as ot
>>> cdf = ot.DistFunc.pNonCentralStudent(2.5, 3.5, 9.5)
>>> cdf = ot.DistFunc.pNonCentralStudent(2.5, 3.5, 9.5, True)