Incomplete beta function

incbet(a, b, x)

Returns incomplete beta integral of the arguments, evaluated from zero to x. The function is defined as

  • a (float) – a positive number
  • b (float) – a positive number
  • x (float) – any number within [0, 1]

See also incbi().


\[\frac{\Gamma(a+b)}{\Gamma(a)+\Gamma(b)} \int_0^xt^{a-1}(1-t)^{b-1} dt\]

The domain of definition is 0 <= x <= 1. In this implementation a and b are restricted to positive values. The integral from x to 1 may be obtained by the symmetry relation:

1 - incbet(a, b, x)  =  incbet(b, a, 1 - x)

The integral is evaluated by a continued fraction expansion or, when b*x is small, by a power series.


Tested at uniformly distributed random points (a, b, x) with a and b in “domain” and x between 0 and 1.

  Relative error
arithmetic domain # trials peak rms
IEEE 0,5 10000 6.9e-15 4.5e-16
IEEE 0,85 250000 2.2e-13 1.7e-14
IEEE 0,1000 30000 5.3e-12 6.3e-13
IEEE 0,10000 250000 9.3e-11 7.1e-12
IEEE 0,100000 10000 8.7e-10 4.8e-11

Outputs smaller than the IEEE gradual underflow threshold were excluded from these statistics.

Error messages

message condition value returned
incbet domain x < 0, x > 1 0.0
incbet underflow 0.0