Incomplete Beta Approximation
Evaluate lower incomplete B_x(a,b) and regularized I_x using symmetry-aware continued fractions, separating normalization; shapes (0,1000], x∈[0,1], bounded iteration with failure refusal.
Description
The incomplete beta integral and its regularized fraction use different normalization.
The incomplete beta integral and its regularized fraction use different normalization. Returning both makes that distinction explicit.1
Evaluate lower incomplete B_x(a,b) and regularized I_x using symmetry-aware continued fractions, separating normalization; shapes (0,1000], x∈[0,1], bounded iteration with failure refusal.
When to use Incomplete Beta Approximation
Use this tool to evaluate lower incomplete B_x(a,b) and regularized I_x using symmetry-aware continued fractions, separating normalization; shapes (0,1000], x∈[0,1], bounded iteration with failure refusal.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Incomplete Beta Approximation and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter a, b, x using the field table and calculation rules, then run the tool.
- Read regularizedLowerBetaFraction, lowerIncompleteBetaValue. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| a | A | number | Required |
| b | B | number | Required |
| x | X | number | Required |
Use positive shapes a,b≤1000 and x in [0,1]. Compute normalization from log gamma and evaluate a symmetry-aware continued fraction, capped at 10,000 iterations with relative criterion 10⁻¹⁴. Return the fraction in [0,1] and the unnormalized integral; endpoints have fraction 0 or 1.
B_x(a,b)=B(a,b)I_x(a,b) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| regularizedLowerBetaFraction | Regularized lower beta fraction | number |
| lowerIncompleteBetaValue | Lower incomplete beta value | number |
{
"input": {
"a": 2,
"b": 3,
"x": 0.5
},
"output": {
"regularizedLowerBetaFraction": 0.6875000000000009,
"lowerIncompleteBetaValue": 0.05729166666666689
}
}Limitations and assumptions
Iteration failure is rejected. Very small unnormalized integrals can underflow; no certified total-error bound is supplied.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Beta Function Approximation: Evaluate positive real beta function and stable logarithm from Lanczos log gamma; no probability distribution sampling or continuation across beta poles. For a complementary task, use Lower Incomplete Gamma Approximation: Evaluate lower integral γ(a,x) and P separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.
References
-
Incomplete Beta Approximation — tool.io
Similar or alternative tools
- Beta Function Approximation
Evaluate positive real beta function and stable logarithm from Lanczos log gamma; no probability distribution sampling or continuation across beta poles.
- Lower Incomplete Gamma Approximation
Evaluate lower integral γ(a,x) and P separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.