# scipy.special.fdtri

scipy.special.fdtri(dfn, dfd, p, out=None) = <ufunc 'fdtri'>

  The p-th quantile of the F-distribution.

  This function is the inverse of the F-distribution CDF, `fdtr`, returning the x such that fdtr(dfn, dfd, x) = p.

  Parameters:
    dfn array_like

      First parameter (positive float).

    dfd array_like

      Second parameter (positive float).

    p array_like

      Cumulative probability, in [0, 1].

    out ndarray, optional

      Optional output array for the function values

  Returns:
    x scalar or ndarray

      The quantile corresponding to p.

  See also

  `fdtr`

    F distribution cumulative distribution function

  `fdtrc`

    F distribution survival function

  `scipy.stats.f`

    F distribution

  Notes

  The computation is carried out using the relation to the inverse regularized beta function, \(I^{-1}_x(a, b)\). Let \(z = I^{-1}_p(d_d/2, d_n/2).\) Then,

  \[x = \frac{d_d (1 - z)}{d_n z}.\]

  If p is such that \(x < 0.5\), the following relation is used instead for improved stability: let \(z' = I^{-1}_{1 - p}(d_n/2, d_d/2).\) Then,

  \[x = \frac{d_d z'}{d_n (1 - z')}.\]

  Wrapper for the Cephes [1] routine `fdtri`.

  The F distribution is also available as `scipy.stats.f`. Calling `fdtri` directly can improve performance compared to the `ppf` method of `scipy.stats.f` (see last example below).

  References

  [1]

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

  Examples

  `fdtri` represents the inverse of the F distribution CDF which is available as `fdtr`. Here, we calculate the CDF for `df1=1`, `df2=2` at `x=3`. `fdtri` then returns `3` given the same values for df1, df2 and the computed CDF value.

      >>> import numpy as np
      >>> from scipy.special import fdtri, fdtr
      >>> df1, df2 = 1, 2
      >>> x = 3
      >>> cdf_value =  fdtr(df1, df2, x)
      >>> fdtri(df1, df2, cdf_value)
      3.000000000000006

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

      >>> x = np.array([0.1, 0.4, 0.7])
      >>> fdtri(1, 2, x)
      array([0.02020202, 0.38095238, 1.92156863])

  Plot the function for several parameter sets.

      >>> import matplotlib.pyplot as plt
      >>> dfn_parameters = [50, 10, 1, 50]
      >>> dfd_parameters = [0.5, 1, 1, 5]
      >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
      >>> parameters_list = list(zip(dfn_parameters, dfd_parameters,
      ...                            linestyles))
      >>> x = np.linspace(0, 1, 1000)
      >>> fig, ax = plt.subplots()
      >>> for parameter_set in parameters_list:
      ...     dfn, dfd, style = parameter_set
      ...     fdtri_vals = fdtri(dfn, dfd, x)
      ...     ax.plot(x, fdtri_vals, label=rf"$d_n={dfn},\, d_d={dfd}$",
      ...             ls=style)
      >>> ax.legend()
      >>> ax.set_xlabel("$x$")
      >>> title = "F distribution inverse cumulative distribution function"
      >>> ax.set_title(title)
      >>> ax.set_ylim(0, 30)
      >>> plt.show()

  The F distribution is also available as `scipy.stats.f`. Using `fdtri` directly can be much faster than calling the `ppf` 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).ppf(x)=fdtri(dfn, dfd, x)`.

      >>> from scipy.stats import f
      >>> dfn, dfd = 1, 2
      >>> x = 0.7
      >>> fdtri_res = fdtri(dfn, dfd, x)  # this will often be faster than below
      >>> f_dist_res = f(dfn, dfd).ppf(x)
      >>> f_dist_res == fdtri_res  # test that results are equal
      True