BETAINV

What it does

BETAINV(probability, alpha, beta, [A], [B])

BETAINV(probability, alpha, beta, [A], [B]) finds a value whose cumulative beta probability is the requested probability. The two positive shape parameters control the distribution, and optional bounds A and B default to zero and one. This is a compatibility name.

Try it

=BETAINV(0.25,1,1,2,6) is about 3. The shape-one beta is uniform on the interval two to six; one quarter of the interval reaches three.

=BETAINV(0.25,2,1,2,6) is about 4. Its standard CDF is z^2, so probability one quarter selects z one half, then 2+4*z=4.

Quirks

Nonpositive shapes and invalid bounds give #NUM!; nonnumeric arguments give #VALUE!. Microsoft’s legacy page excludes probability zero and admits one.

The current shared helper’s validation accepts [0,1], but that is not a promise that probability zero publishes the lower bound in every interval. It has no explicit zero shortcut before its adjacent-value search. On the standard interval this can leave a subnormal result that the current publisher refuses; on two to six, the affine addition rounds the same tiny search value to two. A preview refusal describes the implementation’s limit, rather than a recorded Excel error.

The inverse’s result is a value in the supplied interval, not another probability. A and B are value bounds, while p remains on the probability scale.

How to implement

1. Validate positive finite shapes, finite ordered bounds and the current helper’s probability range. Retain the documented legacy endpoint distinction above.
2. Search z on [0,1] using the current regularized lower beta ratio F(z). The source uses the ordered-binary64 inverse search, with this increasing forward function and target p: bisect the endpoint encodings until adjacent and return the upper one.
3. Keep the callback unchanged. Its interior call supplies bratio(alpha,beta,z,D(1-z)); an independently substituted beta routine changes the surface being inverted.
4. Using ordinary binary64 stores, form width=D(B-A), scaled=D(z*width) and result=D(A+scaled). Apply worksheet publication. Do not contract the final multiplication and addition into an FMA.

With both shapes one, p=0.25 gives z about one quarter, width four and result three. With shapes two and one, F(z)=z^2; z about one half gives result four on the same interval.

For p=0, the current search’s usual strict lower-bracket invariant does not hold: its forward value at zero already equals the target. Without an endpoint shortcut the nonnegative encoding search can terminate at the smallest positive binary64 value. That explains the interval-dependent publication described above; it is a code-path analysis, not an Excel measurement.

A mathematical implementation can handle p=0 explicitly or use a safeguarded Newton/bisection method. Those are different endpoint and stopping policies. Current beta forward errors, stored affine stages and remaining inverse mismatches still need their own comparisons before a whole-function claim.

Rust code

53 betainv "BETAINV" (3, 5) [V "probability", V "alpha", V "beta", V "A", V "B"] |s, a, sv| {
    let [p, alpha, beta] = nums::<3>(s, a)?;
    let lower = if given(a, 3) { num(s, a, 3)? } else { 0.0 };
    let upper = if given(a, 4) { num(s, a, 4)? } else { 1.0 };
    out(s, bg::beta_inv_kernel(p, alpha, beta, lower, upper))
}

No implementation has been published with a Handbook test suite for this function.

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

• XC-BETAINV-001: build 19929 versus build 20131; 2 same-input pairs, 0 differing answers. Compared axes: excel-build. Numeric answers were compared by their recorded binary64 bits; other answers by typed value.

Agreement on these compared inputs does not establish agreement on unmeasured inputs.

Recorded environments and comparison details.

Alternatives

BETA.INV is the current spelling. BETADIST reverses the value/probability direction.

Named variants

No named variants are recorded yet.