# scipy.special.nbdtri

scipy.special.nbdtri(k, n, y, out=None) = <ufunc 'nbdtri'>

  Returns the inverse with respect to the parameter p of y = nbdtr(k, n, p), the negative binomial cumulative distribution function.

  Parameters:
    k array_like

      The maximum number of allowed failures (nonnegative int).

    n array_like

      The target number of successes (positive int).

    y array_like

      The probability of k or fewer failures before n successes (float).

    out ndarray, optional

      Optional output array for the function results

  Returns:
    p scalar or ndarray

      Probability of success in a single event (float) such that nbdtr(k, n, p) = y.

  See also

  `nbdtr`

    Cumulative distribution function of the negative binomial.

  `nbdtrc`

    Negative binomial survival function.

  `scipy.stats.nbinom`

    negative binomial distribution.

  `nbdtrik`

    Inverse with respect to k of nbdtr(k, n, p).

  `nbdtrin`

    Inverse with respect to n of nbdtr(k, n, p).

  `scipy.stats.nbinom`

    Negative binomial distribution

  Notes

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

  The negative binomial distribution is also available as `scipy.stats.nbinom`. Using `nbdtri` directly can improve performance compared to the `ppf` method of `scipy.stats.nbinom`.

  References

  [1]

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

  Examples

  `nbdtri` is the inverse of `nbdtr` with respect to p. Up to floating point errors the following holds: `nbdtri(k, n, nbdtr(k, n, p))=p`.

      >>> import numpy as np
      >>> from scipy.special import nbdtri, nbdtr
      >>> k, n, y = 5, 10, 0.2
      >>> cdf_val = nbdtr(k, n, y)
      >>> nbdtri(k, n, cdf_val)
      0.20000000000000004

  Compute the function for `k=10` and `n=5` at several points by providing a NumPy array or list for y.

      >>> y = np.array([0.1, 0.4, 0.8])
      >>> nbdtri(3, 5, y)
      array([0.34462319, 0.51653095, 0.69677416])

  Plot the function for three different parameter sets.

      >>> import matplotlib.pyplot as plt
      >>> n_parameters = [5, 20, 30, 30]
      >>> k_parameters = [20, 20, 60, 80]
      >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
      >>> parameters_list = list(zip(n_parameters, k_parameters, linestyles))
      >>> cdf_vals = np.linspace(0, 1, 1000)
      >>> fig, ax = plt.subplots(figsize=(8, 8))
      >>> for parameter_set in parameters_list:
      ...     n, k, style = parameter_set
      ...     nbdtri_vals = nbdtri(k, n, cdf_vals)
      ...     ax.plot(cdf_vals, nbdtri_vals, label=rf"$k={k},\ n={n}$",
      ...             ls=style)
      >>> ax.legend()
      >>> ax.set_ylabel("$p$")
      >>> ax.set_xlabel("$CDF$")
      >>> title = "nbdtri: inverse of negative binomial CDF with respect to $p$"
      >>> ax.set_title(title)
      >>> plt.show()

  `nbdtri` can evaluate different parameter sets by providing arrays with shapes compatible for broadcasting for k, n and p. Here we compute the function for three different k at four locations p, resulting in a 3x4 array.

      >>> k = np.array([[5], [10], [15]])
      >>> y = np.array([0.3, 0.5, 0.7, 0.9])
      >>> k.shape, y.shape
      ((3, 1), (4,))

      >>> nbdtri(k, 5, y)
      array([[0.37258157, 0.45169416, 0.53249956, 0.64578407],
             [0.24588501, 0.30451981, 0.36778453, 0.46397088],
             [0.18362101, 0.22966758, 0.28054743, 0.36066188]])