Beta-binomial distribution

Beta-binomial distribution is a type of probability distribution that is derived from the binomial distribution through the assumption of a beta distribution for the probability of success in each trial. It is a compound distribution, meaning that it is constructed from a mixture of two other distributions. This distribution is particularly useful in scenarios where the probability of success is not fixed but can vary according to a beta distribution. It is widely used in Bayesian statistics, epidemiology, and quality control.

Definition

The beta-binomial distribution is defined for a fixed number of trials, \(n\), where each trial has two possible outcomes (commonly referred to as success and failure). The probability of success, \(p\), is not constant but follows a beta distribution characterized by two parameters, \(\alpha\) and \(\beta\). These parameters control the shape of the beta distribution, and consequently, the variability of \(p\) across trials.

The probability mass function (PMF) of the beta-binomial distribution for \(k\) successes out of \(n\) trials is given by:

\[ P(X = k|n, \alpha, \beta) = \binom{n}{k} \frac{B(k + \alpha, n - k + \beta)}{B(\alpha, \beta)} \]

where \(B\) is the beta function, which serves as a normalization constant ensuring that the probabilities sum up to 1.

Properties

The beta-binomial distribution has several important properties:

- **Mean**: The expected number of successes is given by \(E[X] = n\frac{\alpha}{\alpha + \beta}\). - **Variance**: The variance is higher than that of a binomial distribution with the same mean, reflecting the additional uncertainty introduced by the variability in \(p\). It is given by \(Var[X] = n\frac{\alpha\beta(\alpha+\beta+n)}{(\alpha+\beta)^2(\alpha+\beta+1)}\). - **Skewness and Kurtosis**: These measures of the shape of the distribution are also affected by the parameters \(\alpha\) and \(\beta\), indicating the flexibility of the beta-binomial distribution in modeling real-world phenomena where the assumptions of a simple binomial distribution do not hold.

Applications

The beta-binomial distribution finds applications in various fields:

- In Bayesian statistics, it is used as a conjugate prior for the probability of success in binomial experiments. - In epidemiology, it models the distribution of the number of infected individuals in a population where the infection rate varies. - In quality control, it can model the variability in defect rates across different production batches.

Example

Consider a manufacturing process where the probability of producing a defective item is not constant but varies from day to day according to a beta distribution with parameters \(\alpha = 2\) and \(\beta = 5\). If we inspect 10 items, the distribution of the number of defective items follows a beta-binomial distribution with parameters \(n = 10\), \(\alpha = 2\), and \(\beta = 5\).

See Also

• Binomial distribution
• Beta distribution
• Probability distribution
• Bayesian statistics

External Links

• [Beta-binomial distribution on Wikimedia Commons](https://commons.wikimedia.org/wiki/Category:Beta-binomial_distribution)