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Function std.mathspecial.betaIncomplete

Regularized incomplete beta function Ix(a,b)

real betaIncomplete(
  real a,
  real b,
  real x
) pure nothrow @nogc @safe;

Mathematically, if a and b are positive real numbers, and 0 ≤ x ≤ 1, then Ix(a,b) = ∫0xta-1(1-t)b-1dt/B(a,b) where B is the beta function. It is also the cumulative distribution function of the beta distribution.

betaIncomplete(a, b, x) evaluates Ix(a,b).

Parameters

NameDescription
a the first argument of B, must be positive
b the second argument of B, must be positive
x the fraction of integration completion from below, 0 ≤ x ≤ 1

Returns

It returns Ix(a,b), an element of [0,1].

Special Values
a b x betaIncomplete(a, b, x)
negative b x NAN
a negative x NAN
a b < 0 NAN
a b > 1 NAN
+0 +0 (0,1) NAN
∞ ∞ (0,1) NAN

If one or more of the input parameters are NAN, the one with the largest payload is returned. For equal payloads but with possibly different signs, the order of preference is x, a, b.

Note

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

See Also

beta betaIncompleteCompl

Example

writeln(betaIncomplete(1, 1, .5)); // .5
writeln(betaIncomplete(+0., +0., 0)); // 0
assert(isNaN(betaIncomplete(+0., +0., .5)));
assert(isNaN(betaIncomplete(real.infinity, real.infinity, .5)));
writeln(betaIncomplete(real.infinity, real.infinity, 1)); // 1
assert(betaIncomplete(NaN(0x1), 1, NaN(0x2)) is NaN(0x2));
assert(betaIncomplete(1, NaN(0x3), -NaN(0x3)) is -NaN(0x3));

Authors

Stephen L. Moshier (original C code). Conversion to D by Don Clugston

License

Boost License 1.0.