scipy.stats.t

scipy.stats.t = <scipy.stats._continuous_distns.t_gen object>

A Student’s t continuous random variable.

For the noncentral t distribution, see nct.

As an instance of the rv_continuous class, t object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

See also

nct

Notes

The probability density function for t is:

\[f(x, \nu) = \frac{\Gamma((\nu+1)/2)} {\sqrt{\pi \nu} \Gamma(\nu/2)} (1+x^2/\nu)^{-(\nu+1)/2}\]

where \(x\) is a real number and the degrees of freedom parameter \(\nu\) (denoted df in the implementation) satisfies \(\nu > 0\). \(\Gamma\) is the gamma function (scipy.special.gamma).

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, t.pdf(x, df, loc, scale) is identically equivalent to t.pdf(y, df) / scale with y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

Examples

>>> from scipy.stats import t
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> df = 2.74
>>> mean, var, skew, kurt = t.stats(df, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(t.ppf(0.01, df),
...                 t.ppf(0.99, df), 100)
>>> ax.plot(x, t.pdf(x, df),
...        'r-', lw=5, alpha=0.6, label='t pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

>>> rv = t(df)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = t.ppf([0.001, 0.5, 0.999], df)
>>> np.allclose([0.001, 0.5, 0.999], t.cdf(vals, df))
True

Generate random numbers:

>>> r = t.rvs(df, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

Methods

rvs(df, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, df, loc=0, scale=1)	Probability density function.
logpdf(x, df, loc=0, scale=1)	Log of the probability density function.
cdf(x, df, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, df, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, df, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, df, loc=0, scale=1)	Log of the survival function.
ppf(q, df, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, df, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(n, df, loc=0, scale=1)	Non-central moment of order n
stats(df, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(df, loc=0, scale=1)	(Differential) entropy of the RV.
fit(data)	Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
expect(func, args=(df,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)	Expected value of a function (of one argument) with respect to the distribution.
median(df, loc=0, scale=1)	Median of the distribution.
mean(df, loc=0, scale=1)	Mean of the distribution.
var(df, loc=0, scale=1)	Variance of the distribution.
std(df, loc=0, scale=1)	Standard deviation of the distribution.
interval(alpha, df, loc=0, scale=1)	Endpoints of the range that contains fraction alpha [0, 1] of the distribution

scipy.stats.studentized_range scipy.stats.trapezoid