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# scipy.special.btdtr

scipy.special.btdtr(a, b, x, out=None)

  Cumulative distribution function of the beta distribution.

  Returns the integral from zero to x of the beta probability density function,

  \[I = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt\]

  where \(\Gamma\) is the gamma function.

  Deprecated since version 1.12.0: This function is deprecated and will be removed from SciPy 1.14.0. Use `scipy.special.betainc` instead.

  Parameters:
    a array_like

      Shape parameter (a > 0).

    b array_like

      Shape parameter (b > 0).

    x array_like

      Upper limit of integration, in [0, 1].

    out ndarray, optional

      Optional output array for the function values

  Returns:
    I scalar or ndarray

      Cumulative distribution function of the beta distribution with parameters a and b at x.

  See also

  `betainc`

  Notes

  This function is identical to the incomplete beta integral function `betainc`.

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

  References

  [1]

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