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# scipy.special.nctdtr

scipy.special.nctdtr(df, nc, t, out=None) = <ufunc 'nctdtr'>

  Cumulative distribution function of the non-central t distribution.

  Parameters:
    dfarray_like

      Degrees of freedom of the distribution. Should be in range (0, inf).

    ncarray_like

      Noncentrality parameter.

    tarray_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

  `nctdtrit`

    Inverse CDF (iCDF) of the non-central t distribution.

  `nctdtridf`

    Calculate degrees of freedom, given CDF and iCDF values.

  `nctdtrinc`

    Calculate non-centrality parameter, given CDF iCDF values.

  Notes

  This function calculates the CDF of the non-central t distribution using the Boost Math C++ library [1].

  Note that the argument order of `nctdtr` is different from that of the similar `cdf` method of `scipy.stats.nct`: t is the last parameter of `nctdtr` but the first parameter of `scipy.stats.nct.cdf`.

  References

  [1]

  The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

  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 t distribution, for nc=0. Compare with the t-distribution from scipy.stats:

      >>> x = np.linspace(-5, 5, num=500)
      >>> df = 3
      >>> nct_stats = stats.t.cdf(x, df)
      >>> nct_special = special.nctdtr(df, 0, x)

      >>> fig = plt.figure()
      >>> ax = fig.add_subplot(111)
      >>> ax.plot(x, nct_stats, 'b-', lw=3)
      >>> ax.plot(x, nct_special, 'r-')
      >>> plt.show()