scipy.special.fdtr#
- 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 asscipy.stats.f
. Callingfdtr
directly can improve performance compared to thecdf
method ofscipy.stats.f
(see last example below).References
[1]Cephes Mathematical Functions Library, http://www.netlib.org/cephes/
Examples
Calculate the function for
dfn=1
anddfd=2
atx=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
. Usingfdtr
directly can be much faster than calling thecdf
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