# scipy.special.fdtrc

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

  F survival function.

  Returns the complemented F-distribution function (the integral of the density from x to infinity).

  Parameters:
    dfn array_like

      First parameter (positive float).

    dfd array_like

      Second parameter (positive float).

    x array_like

      Argument (nonnegative float).

    out ndarray, optional

      Optional output array for the function values

  Returns:
    y scalar or ndarray

      The complemented F-distribution function with parameters dfn and dfd at x.

  See also

  `fdtr`

    F distribution cumulative distribution function

  `fdtri`

    F distribution inverse cumulative distribution function

  `scipy.stats.f`

    F distribution

  Notes

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

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

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

  References

  [1]

  Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

  Examples

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

      >>> import numpy as np
      >>> from scipy.special import fdtrc
      >>> fdtrc(1, 2, 1)
      0.42264973081037427

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

      >>> x = np.array([0.5, 2., 3.])
      >>> fdtrc(1, 2, x)
      array([0.5527864 , 0.29289322, 0.22540333])

  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
      ...     fdtrc_vals = fdtrc(dfn, dfd, x)
      ...     ax.plot(x, fdtrc_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
      ...             ls=style)
      >>> ax.legend()
      >>> ax.set_xlabel("$x$")
      >>> ax.set_title("F distribution survival function")
      >>> plt.show()

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

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