scipy.stats.f — SciPy v1.15.3 Manual (original) (raw)

scipy.stats.f = <scipy.stats._continuous_distns.f_gen object>[source]#

An F continuous random variable.

For the noncentral F distribution, see ncf.

As an instance of the rv_continuous class, f 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.

Notes

The F distribution with \(df_1 > 0\) and \(df_2 > 0\) degrees of freedom is the distribution of the ratio of two independent chi-squared distributions with\(df_1\) and \(df_2\) degrees of freedom, after rescaling by\(df_2 / df_1\).

The probability density function for f is:

\[f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}} {(df_2+df_1 x)^{(df_1+df_2)/2} B(df_1/2, df_2/2)}\]

for \(x > 0\).

f accepts shape parameters dfn and dfd for \(df_1\), the degrees of freedom of the chi-squared distribution in the numerator, and \(df_2\), the degrees of freedom of the chi-squared distribution in the denominator, respectively.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, f.pdf(x, dfn, dfd, loc, scale) is identically equivalent to f.pdf(y, dfn, dfd) / scale withy = (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

import numpy as np from scipy.stats import f import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

dfn, dfd = 29, 18 mean, var, skew, kurt = f.stats(dfn, dfd, moments='mvsk')

Display the probability density function (pdf):

x = np.linspace(f.ppf(0.01, dfn, dfd), ... f.ppf(0.99, dfn, dfd), 100) ax.plot(x, f.pdf(x, dfn, dfd), ... 'r-', lw=5, alpha=0.6, label='f 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 = f(dfn, dfd) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

vals = f.ppf([0.001, 0.5, 0.999], dfn, dfd) np.allclose([0.001, 0.5, 0.999], f.cdf(vals, dfn, dfd)) True

Generate random numbers:

r = f.rvs(dfn, dfd, size=1000)

And compare the histogram:

ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) ax.set_xlim([x[0], x[-1]]) ax.legend(loc='best', frameon=False) plt.show()

Methods