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What is the formula for the incomplete beta function?

Published in Incomplete Beta Function 3 mins read

The formula for the incomplete beta function is primarily defined through an integral. There are also important variations and series expansions used in practical applications.

The incomplete beta function, denoted as $B_x(a,b)$, is defined by the integral:

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

Where:

  • $a$ and $b$ are positive real numbers, often referred to as shape parameters.
  • $x$ is the upper limit of integration, constrained to $0 \le x \le 1$.
  • $t$ is the integration variable.

The Regularized Incomplete Beta Function

Often, when people refer to the "incomplete beta function" in statistical and computational contexts, they are actually referring to the regularized incomplete beta function, denoted as $I_x(a,b)$. This is the incomplete beta function normalized by the complete beta function $B(a,b)$, which is the same integral evaluated from 0 to 1:

$$B(a,b) = \int_0^1 t^{a-1}(1-t)^{b-1} dt$$

The formula for the regularized incomplete beta function is:

$$I_x(a,b) = \frac{B_x(a,b)}{B(a,b)} = \frac{\int_0^x t^{a-1}(1-t)^{b-1} dt}{\int_0^1 t^{a-1}(1-t)^{b-1} dt}$$

The regularized incomplete beta function is particularly significant because it represents the cumulative distribution function (CDF) of the Beta distribution. This means $I_x(a,b)$ gives the probability that a Beta-distributed random variable with parameters $a$ and $b$ will take a value less than or equal to $x$.

Series Expansion Formula

For computational purposes, especially when the parameter $a$ is an integer, the regularized incomplete beta function can also be expressed as a finite series. One important series expansion for $I_x(a,b)$ is:

$$Ix(a,b) = (1-x)^{a+b-1} \sum{i=0}^{a-1} \binom{a+b-1}{i} \left(\frac{x}{1-x}\right)^i$$

This formula is particularly useful when $a$ is an integer, and it highlights the connection between the Beta distribution and the Binomial distribution's cumulative probabilities.

Key Parameters Explained

To better understand these formulas, let's break down the parameters:

  • a (Alpha Parameter): This parameter influences the shape of the Beta distribution. A larger a value typically shifts the peak of the distribution towards 1.
  • b (Beta Parameter): Similar to a, this parameter also shapes the distribution. A larger b value tends to shift the peak towards 0.
  • x (Upper Limit): This value, ranging from 0 to 1, defines the point up to which the integral is evaluated. In a probabilistic context, it represents the specific threshold for which the cumulative probability is being calculated.

Applications

The incomplete beta function and its regularized form are fundamental in various fields, including:

  • Statistics and Probability: Essential for working with the Beta distribution, which models probabilities, proportions, or rates.
  • Hypothesis Testing: Used in tests involving binomial probabilities and order statistics.
  • Machine Learning: Employed in Bayesian inference, especially when dealing with prior distributions and likelihoods that can be modeled by Beta distributions.
  • Engineering and Finance: Applicable in scenarios requiring the modeling of random variables bounded between 0 and 1, such as success rates or project completion percentages.

Understanding these formulas is crucial for anyone working with continuous probability distributions and their applications.