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Incomplete Beta Calculator

Incomplete beta B_x(a, b) = ∫₀ˣ t^{a−1}(1−t)^{b−1} dt for a>0, b>0, 0≤x≤1. Same engine as Special Functions. Not normalized sinc. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 15/15 declared partitions (erf, erfc, gamma, lgamma, bessel-j, bessel-y, bessel-i, bessel-k, lambert-w, lambert-wm1, beta, incomplete-beta, sinc, normalized-sinc, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Incomplete beta B_x(a, b) for a>0, b>0, 0≤x≤1, with an engine-linked Live graph vs x.
Scope
Incomplete Euler beta
Verification
Engine tested · Source checked · v1.10.2 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.10.2 · CVP protocol 1.0.0-proposed · Evidence 2026-10-08.math-response-consistency
Verification revision
2026-10-08.math-response-consistency · 1/1 property · digest 0b2836762d99
Legacy regression
58/58 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Special response contract also checks all 14 mode formulas and unrounded numeric summaries, including erfc/K tails and a subnormal erf input; UI is shared presentation/SSR, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
482/482 golden · 8/8 CVP boundary · 10/10 invalid · 1/1 property · 3/3 metamorphic · 24/24 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
13 legacy golden · 8 legacy boundary · legacy regression suite · 482/482 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.10.2 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

DefinitionB_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt
Complete at x=1B_1(a, b) = B(a, b)
iSame engine as /special-functions. Not a second engine. Not /ibeta. Normalized sinc is on /normalized-sinc.

How to use

1

Enter x, a, and b

Finite reals with a>0, b>0, and 0≤x≤1. Defaults x=0.5, a=2, b=3.

2

Read B_x(a, b)

B_{0.5}(2, 3) ≈ 0.0572916666667. B_1(2, 3)=1/12. Regularized I_x is B_x/B. The Live graph is y = B_x(a, b) on [0, 1] with a, b held.

Example calculations

Common configurations with formula and result.

ϟ

B_{0.5}(2, 3)

x = 0.5, a = 2, b = 3

B_{0.5}(2, 3)
0.0572916666667
ϟ

B_{0.5}(1/2, 1/2)

x = 0.5, a = 0.5, b = 0.5

B_{1/2}(1/2, 1/2) = π/2
π/2

Incomplete Beta calculator specification

Version 1.10.2 · Engine tested

Calculation status

Review policy · Evidence

Definition
The incomplete beta B_x(a, b) = ∫₀ˣ t^{a−1}(1−t)^{b−1} dt for a>0, b>0, and 0≤x≤1. B_0=0, B_1(a,b)=B(a,b), B_x(1,1)=x, B_{1/2}(1/2,1/2)=π/2. Regularized I_x(a,b)=B_x/B is returned as regularized. This page is mode=incomplete_beta on math.special. Not normalized sinc. Not CAS.
What it calculates
Incomplete beta B_x(a, b) for a>0, b>0, 0≤x≤1, with an engine-linked Live graph vs x.
Inputs
  • x
  • a
  • b
Outputs
  • value
  • regularized
  • x
  • a
  • b
Formula
B_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt
Assumptions
  • Incomplete Euler beta
  • a>0, b>0, 0≤x≤1
  • The Result Card includes an engine-linked Live graph of B_x(a, b) vs x on [0, 1] with a, b held. Same series as REST/MCP. Not normalized sinc. Not CAS.
Units
  • dimensionless
Boundary conditions
  • non-finite x, a, or b → INVALID_NUMBER
  • a ≤ 0 or b ≤ 0 → INVALID_INPUT
  • x < 0 or x > 1 → INVALID_INPUT
  • unknown mode (normalized sinc / CAS) → INVALID_MODE
Example
x=0.5 a=2 b=3 → 0.0572916666667
Validation cases

2 published on this page · 58/58 tests · Production surface contract 6/6 · View evidence

  • x=0.5 a=2 b=3 → 0.0572916666667
  • x=0.5 a=0.5 b=0.5 → π/2
Sources
Calculation version
1.10.2

Background

Interpretation and common distinctions.

What does this incomplete beta calculator compute?

The incomplete beta B_x(a, b) = ∫₀ˣ t^{a−1}(1−t)^{b−1} dt for a>0, b>0, and 0≤x≤1. B_0=0, B_1(a,b)=B(a,b), B_x(1,1)=x, B_{1/2}(1/2,1/2)=π/2. Regularized I_x(a,b)=B_x/B is returned as regularized.

The published formula or method is B_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt. The following references support the stated mathematics or method:

Which inputs and limits should you check?

The mathematical function can have a wider domain than this calculator supports. Choose the intended function and branch, then check its input domain and published precision limits.

The worked example uses x = 0.5, a = 2, b = 3 with B_{0.5}(2, 3). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.special v1.10.2 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel K Calculator, Bessel Y Calculator, Beta Function Calculator, Error Function Calculator, Gamma Function Calculator, Lambert W Calculator . Explore all Special Functions.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator Calculate the Bessel function J_n(x) for integer orders 0–8 and real |x|≤80. See the formula, graph, numerical method, references, and versioned evidence. › → Bessel K Calculator K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → Error Function Calculator erf(x) = (2/√π) ∫₀ˣ e^{−t²} dt. Same engine as Special Functions. Odd, erf(0)=0, erf(∞)→1. Not CAS. Runs locally. › → Gamma Function Calculator Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt except poles at 0, −1, −2, …. Same engine as Special Functions. Not factorial. Not CAS. Runs locally. › → Lambert W Calculator Principal real W_0(x) for x ≥ −1/e. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second special-functions engine?

No. It is mode=incomplete_beta on math.special, the same engine as /calc/math/special-functions.

Is this normalized sinc?

No. This seed is the incomplete beta B_x(a,b) for a>0, b>0, 0≤x≤1. Complete Euler B is on /beta-function. Normalized sinc is on /normalized-sinc. Not CAS.

Where does this run?

Locally in the browser by default. REST and MCP call the same special-functions engine.