1. Introduction

The Lower Incomplete Beta Function Calculator computes the function \( B_x(\alpha, \beta) \) for parameters \( \alpha \) (alpha), \( \beta \) (beta), and \( x \) (0 ≤ x ≤ 1). This function is vital for statistical analysis, including Bayesian inference and hypothesis testing.

2. What is the Lower Incomplete Beta Function?

The function \( B_x(\alpha, \beta) \) is defined as:

$$B_x(\alpha, \beta) = \int_0^x t^{\alpha - 1} (1 - t)^{\beta - 1} dt$$

where:

• α (Alpha): First shape parameter.
• β (Beta): Second shape parameter.
• x: The upper limit of integration (0 ≤ x ≤ 1).

This function is essential for calculating probabilities related to the Beta distribution.

3. Features of the Calculator

• User-Friendly Interface: Clean and intuitive design.
• Input Parameters: Enter α, β, and x (0 ≤ x ≤ 1).
• Accurate Computations: Uses Lanczos approximation for Gamma function and the continued fraction method for accuracy.
• Error Handling: Provides clear messages for invalid entries.
• Responsive Design: Compatible across desktops, tablets, and smartphones.
• Clear Display: Shows results and their interpretation.
• Interactive Example: Demonstrates the calculator's functionality.

4. How the Calculator Works

Steps for computing \( B_x(\alpha, \beta) \):

1. Input Collection: Enter values for α, β, and x.
2. Validation: Ensure α and β are positive, and x is between 0 and 1.
3. Computation:
   • Compute Gamma function values using Lanczos approximation.
   • Calculate Beta function \( B(\alpha, \beta) \).
   • Compute the Regularized Lower Incomplete Beta Function \( I_x(\alpha, \beta) \).
   • Derive \( B_x(\alpha, \beta) \) by multiplying \( I_x(\alpha, \beta) \) with \( B(\alpha, \beta) \).
4. Result Display: Shows values and interpretations.

5. Step-by-Step Guide

1. Open the Calculator:
   • Save the file as lower_incomplete_beta_calculator.html.
   • Open in a browser (e.g., Chrome, Firefox).
2. Input Parameters:
   • α: Enter a positive value.
   • β: Enter a positive value.
   • x: Enter a value between 0 and 1.
3. Click "Compute" to calculate.
4. Review Results: Displays values and their interpretation.

6. Practical Example

7. Additional Notes

• Importance of \( B_x(\alpha, \beta) \) in statistics.
• Explanation of the Gamma and Beta functions.
• Applications in Bayesian statistics, hypothesis testing, and probability modeling.
• Edge cases and assumptions for α, β, and x.

8. Frequently Asked Questions (FAQ)

Q1: What does \( B_x(\alpha, \beta) \) represent?

A: The integral of the Beta distribution's PDF up to \( x \).

Q2: Can α and β be non-integers?

A: Yes, they can be any positive real values.

Q3: What if \( x = 0 \) or \( x = 1 \)?

A: \( B_x(\alpha, \beta) = 0 \) for \( x = 0 \), and \( B_x(\alpha, \beta) = B(\alpha, \beta) \) for \( x = 1 \).