scipy.special.fdtr — SciPy v1.15.2 Manual (original) (raw)

scipy.special.fdtr(dfn, dfd, x, out=None) = <ufunc 'fdtr'>#

F cumulative distribution function.

Returns the value of the cumulative distribution function of the F-distribution, also known as Snedecor’s F-distribution or the Fisher-Snedecor distribution.

The F-distribution with parameters \(d_n\) and \(d_d\) is the distribution of the random variable,

\[X = \frac{U_n/d_n}{U_d/d_d},\]

where \(U_n\) and \(U_d\) are random variables distributed\(\chi^2\), with \(d_n\) and \(d_d\) degrees of freedom, respectively.

Parameters:

dfnarray_like

First parameter (positive float).

dfdarray_like

Second parameter (positive float).

xarray_like

Argument (nonnegative float).

outndarray, optional

Optional output array for the function values

Returns:

yscalar or ndarray

The CDF of the F-distribution with parameters dfn and dfd at x.

See also

fdtrc

F distribution survival function

fdtri

F distribution inverse cumulative distribution

scipy.stats.f

F distribution

Notes

The regularized incomplete beta function is used, according to the formula,

\[F(d_n, d_d; x) = I_{xd_n/(d_d + xd_n)}(d_n/2, d_d/2).\]

Wrapper for the Cephes [1] routine fdtr. The F distribution is also available as scipy.stats.f. Calling fdtr directly can improve performance compared to the cdf method of scipy.stats.f (see last example below).

References

Examples

Calculate the function for dfn=1 and dfd=2 at x=1.

import numpy as np from scipy.special import fdtr fdtr(1, 2, 1) 0.5773502691896258

Calculate the function at several points by providing a NumPy array for_x_.

x = np.array([0.5, 2., 3.]) fdtr(1, 2, x) array([0.4472136 , 0.70710678, 0.77459667])

Plot the function for several parameter sets.

import matplotlib.pyplot as plt dfn_parameters = [1, 5, 10, 50] dfd_parameters = [1, 1, 2, 3] linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] parameters_list = list(zip(dfn_parameters, dfd_parameters, ... linestyles)) x = np.linspace(0, 30, 1000) fig, ax = plt.subplots() for parameter_set in parameters_list: ... dfn, dfd, style = parameter_set ... fdtr_vals = fdtr(dfn, dfd, x) ... ax.plot(x, fdtr_vals, label=rf"$d_n={dfn},, d_d={dfd}$", ... ls=style) ax.legend() ax.set_xlabel("$x$") ax.set_title("F distribution cumulative distribution function") plt.show()

The F distribution is also available as scipy.stats.f. Using fdtrdirectly can be much faster than calling the cdf method ofscipy.stats.f, especially for small arrays or individual values. To get the same results one must use the following parametrization:stats.f(dfn, dfd).cdf(x)=fdtr(dfn, dfd, x).

from scipy.stats import f dfn, dfd = 1, 2 x = 1 fdtr_res = fdtr(dfn, dfd, x) # this will often be faster than below f_dist_res = f(dfn, dfd).cdf(x) fdtr_res == f_dist_res # test that results are equal True