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Beta Function

What is the Beta Function?

The Beta function \(B(x, y)\) is a continuous function that tells you how to weight values between 0 and 1 when combining two quantities with different “pulls” or “strengths” \(x\) and \(y\). The Beta function takes two inputs because it measures interactions between two quantities, it’s about how two competing forces shape a distribution over the interval \([0,1]\). It is also closely related to the Gamma function.

For example:

  • \(x\) controls how strongly we pull toward 1
  • \(y\) controls how strongly we pull toward 0

It acts like a balancing tool:

  • When you want to measure how much a value leans toward one side (e.g., near 0 or near 1),
  • And you want both sides to influence the outcome smoothly.

Interpretation

  • The Beta function acts like a normalization constant for the Beta distribution.
  • It provides a way to combine two inputs \(x\) and \(y\) in a symmetric and continuous fashion.
  • It's useful in modeling probabilities over intervals (e.g., proportions, rates, percentages between 0 and 1).

Definition

For real numbers \(x > 0\) and \(y > 0\), the Beta function \(B(x, y)\) is defined as:

\[ B(x, y) = \int_0^1 t^{x - 1} (1 - t)^{y - 1} \, dt \]

This integral converges for all positive \(x\) and \(y\).

Properties

Property Description
Symmetry \(B(x, y) = B(y, x)\)
Relation to Gamma \(B(x, y) = \dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x + y)}\)
Positive \(B(x, y) > 0\) for \(x, y > 0\)
Used in distributions Normalizes the PDF of the Beta distribution
Special values \(B(1, y) = \frac{1}{y}\), \(B(x, 1) = \frac{1}{x}\)

Why is the Beta Function important in probability?

  • It normalizes the Beta distribution, making the area under the PDF equal to 1.
  • It models random variables bounded between 0 and 1 (e.g., proportions).
  • Its relationship with the Gamma function helps simplify many statistical formulas.

Connection to Gamma Function

The Beta function can be expressed using the Gamma function:

\[ B(x, y) = \frac{\Gamma(x) \cdot \Gamma(y)}{\Gamma(x + y)} \]

This identity is extremely useful when computing Beta-related expressions in probability or simplifying integrals.

Example

Let’s compute \(B(2, 3)\):

Using the definition:

\[ \begin{align*}B(2, 3) = \int_0^1 t^{1} (1 - t)^{2} \, dt = \int_0^1 t (1 - t)^2 \, dt\\= \int_0^1 t (1 - 2t + t^2) \, dt = \int_0^1 (t - 2t^2 + t^3) \, dt\\= \left[ \frac{t^2}{2} - \frac{2t^3}{3} + \frac{t^4}{4} \right]_0^1 = \frac{1}{2} - \frac{2}{3} + \frac{1}{4}\\= \frac{6}{12} - \frac{8}{12} + \frac{3}{12} = \frac{6 - 8 + 3}{12} = \frac{1}{12}\end{align*} \]

Using the Gamma relation:

\[ B(2, 3) = \frac{\Gamma(2)\Gamma(3)}{\Gamma(5)} = \frac{1! \cdot 2!}{4!} = \frac{1 \cdot 2}{24} = \frac{1}{12} \]

Use in Beta Distribution

The Beta distribution with parameters \(\alpha > 0\), \(\beta > 0\) has PDF:

\[ f(x; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, \quad \text{for } x \in (0, 1) \]

Here, the Beta function in the denominator ensures that the area under the PDF is 1.