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# scipy.special.nbdtrin

scipy.special.nbdtrin(k, y, p, out=None) = <ufunc 'nbdtrin'>

  Inverse of `nbdtr` vs n.

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

  Parameters:
    k array_like

      The maximum number of allowed failures (nonnegative int).

    y array_like

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

    p array_like

      Probability of success in a single event (float).

    out ndarray, optional

      Optional output array for the function results

  Returns:
    n scalar or ndarray

      The number of successes n such that nbdtr(k, n, p) = y.

  See also

  `nbdtr`

    Cumulative distribution function of the negative binomial.

  `nbdtri`

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

  `nbdtrik`

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

  Notes

  This function wraps routines from the Boost Math C++ library [1].

  Formula 26.5.26 of [2] or [3],

  \[\sum_{j=k + 1}^\infty {{n + j - 1} \choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),\]

  is used to reduce calculation of the cumulative distribution function to that of a regularized incomplete beta \(I\).

  Computation of n involves a search for a value that produces the desired value of y. The search relies on the monotonicity of y with n.

  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.

  [3]

  NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/8.17.E24

  Examples

  Compute the negative binomial cumulative distribution function for an exemplary parameter set.

      >>> from scipy.special import nbdtr, nbdtrin
      >>> k, n, p = 5, 2, 0.5
      >>> cdf_value = nbdtr(k, n, p)
      >>> cdf_value
      0.9375

  Verify that `nbdtrin` recovers the original value for n up to floating point accuracy.

      >>> nbdtrin(k, cdf_value, p)
      2.0