1. Introduction

The Beta Distribution PDF Calculator computes the Probability Density Function (PDF) of a Beta-distributed variable. It is useful in statistics for modeling probabilities, proportions, and rates.

2. What is the Beta Distribution?

The Beta distribution is defined on the interval [0, 1] and parameterized by α (alpha) and β (beta). It is widely used in Bayesian statistics and other applications where values are restricted to finite intervals.

Probability Density Function (PDF):

$$ f(x; \alpha, \beta) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)} $$

• α (Alpha): First shape parameter.
• β (Beta): Second shape parameter.
• B(α, β): Beta function, defined as:

$$ B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} $$

where Γ(z) is the Gamma function.

3. Features of the Beta PDF Calculator

• User-Friendly Interface: Clean design suitable for both desktops and mobile devices.
• Input Parameters: Enter α, β, and x (0 ≤ x ≤ 1).
• Accurate Computations: Uses Lanczos approximation for precise calculations.
• Error Handling: Validates inputs for acceptable ranges and formats.
• Clear Results Display: Shows calculated PDF value and its interpretation.
• Interactive Example: Demonstrates functionality with a practical example.

4. How the Calculator Works

The Beta PDF Calculator computes \( f(x; \alpha, \beta) \) as follows:

1. Input Collection: Users enter values for α, β, and x.
2. Validation: Ensures α and β are positive, and x is within [0, 1].
3. Computation:
   • Calculates Beta function using the Gamma function.
   • Applies the PDF formula to compute \( f(x; \alpha, \beta) \).
4. Result Display: Shows the computed PDF value and interpretation.

5. Step-by-Step Guide to Using the Calculator

1. Open the Calculator:
   • Save as beta_pdf_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 PDF".
4. Review Results: Displays the PDF value and interpretation.

6. Practical Example

7. Additional Notes

• The PDF value at \( x \) gives the relative likelihood, not a probability.
• Uses Lanczos approximation for Gamma function calculations.
• α and β must be positive; x must be within [0, 1].
• Applications include Bayesian statistics, quality control, and more.
• Edge cases like uniform distribution when \( \alpha = 1 \) and \( \beta = 1 \).

8. Frequently Asked Questions (FAQ)

What does the PDF value represent?

It indicates the density or relative likelihood of the random variable being at that point.

Can I input non-integer values for α and β?

Yes, they can be any positive real values.

What happens if I input \( x = 0 \) or \( x = 1 \)?

Depends on α and β values:

• If \( \alpha < 1 \) or \( \beta < 1 \), PDF tends to infinity.
• If \( \alpha \geq 1 \) and \( \beta \geq 1 \), PDF is zero.