scipy.special.ncfdtr#
- scipy.special.ncfdtr(dfn, dfd, nc, f, out=None) = <ufunc 'ncfdtr'>#
- Cumulative distribution function of the non-central F distribution. - The non-central F describes the distribution of, \[Z = \frac{X/d_n}{Y/d_d}\]- where \(X\) and \(Y\) are independently distributed, with \(X\) distributed non-central \(\chi^2\) with noncentrality parameter nc and \(d_n\) degrees of freedom, and \(Y\) distributed \(\chi^2\) with \(d_d\) degrees of freedom. - Parameters:
- dfnarray_like
- Degrees of freedom of the numerator sum of squares. Range (0, inf). 
- dfdarray_like
- Degrees of freedom of the denominator sum of squares. Range (0, inf). 
- ncarray_like
- Noncentrality parameter. Range [0, inf). 
- farray_like
- Quantiles, i.e. the upper limit of integration. 
- outndarray, optional
- Optional output array for the function results 
 
- Returns:
- cdfscalar or ndarray
- The calculated CDF. If all inputs are scalar, the return will be a float. Otherwise it will be an array. 
 
 - See also - ncfdtri
- Quantile function; inverse of - ncfdtrwith respect to f.
- ncfdtridfd
- Inverse of - ncfdtrwith respect to dfd.
- ncfdtridfn
- Inverse of - ncfdtrwith respect to dfn.
- ncfdtrinc
- Inverse of - ncfdtrwith respect to nc.
- scipy.stats.ncf
- Non-central F distribution. 
 - Notes - This function calculates the CDF of the non-central f distribution using the Boost Math C++ library [1]. - The cumulative distribution function is computed using Formula 26.6.20 of [2]: \[F(d_n, d_d, n_c, f) = \sum_{j=0}^\infty e^{-n_c/2} \frac{(n_c/2)^j}{j!} I_{x}(\frac{d_n}{2} + j, \frac{d_d}{2}),\]- where \(I\) is the regularized incomplete beta function, and \(x = f d_n/(f d_n + d_d)\). - Note that argument order of - ncfdtris different from that of the similar- cdfmethod of- scipy.stats.ncf: f is the last parameter of- ncfdtrbut the first parameter of- scipy.stats.ncf.cdf.- References [1]- The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/. [2]- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. - Examples - >>> import numpy as np >>> from scipy import special >>> from scipy import stats >>> import matplotlib.pyplot as plt - Plot the CDF of the non-central F distribution, for nc=0. Compare with the F-distribution from scipy.stats: - >>> x = np.linspace(-1, 8, num=500) >>> dfn = 3 >>> dfd = 2 >>> ncf_stats = stats.f.cdf(x, dfn, dfd) >>> ncf_special = special.ncfdtr(dfn, dfd, 0, x) - >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> ax.plot(x, ncf_stats, 'b-', lw=3) >>> ax.plot(x, ncf_special, 'r-') >>> plt.show() 