# Beta

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## Revision as of 19:41, 16 March 2010

## Contents |

## The Beta Distribution

- The Beta Distribution can be used for a continuous random variable that can only take values between 0 and 1.
- Can be used to describe the amount of time an event or project will take to complete, if it must take place within a specified interval.

## Probability Density Function

- The leading term (with the Gamma functions) is a constant that ensures the area under the pdf curve = 1.
- R code: dbeta(p,alpha,beta), where the sum alpha+beta is regarded as the "sample size" and alpha/(alpha+beta) is the population mean.
- Another useful parameterization is a = alpha/(alpha+beta) and b = 1/sqrt(alpha+beta)
- alpha = a/b^2
- beta = (1-a)/b^2

## Cumulative Distribution Function

- where Bx(α,β) is the incomplete beta function and Ix(α,β) is the regularized incomplete beta function.
- R code: pbeta(q, alpha, beta)

## Generating Random Variables

> rbeta(10,5,8)

[1] 0.28396857 0.29257510 0.43823168 0.46036009 0.31662962 0.31494870 [7] 0.31709740 0.35495964 0.39718446 0.07957405

## Common Statistics

## Parameter Estimation

### Bayesian

Two cases are described below. In the first, we are given a sample drawn directly from a beta-distributed population. In the second, rathe than observing the sampled probabilities directly, we're given the results of repeated Bernoulli trials (i.e., Binomial trials, with perhaps-different orders (Ns)) for each unobserved probability.

#### Data = Beta-Distributed Probabilities = c(p1, p2, ...pn)

The likelihood function is simply the product of beta densities.

- L(data|alpha,beta) = dbeta(p1,alpha,beta)*dbeta(p2,alpha,beta)*...*dbeta(pn,alpha,beta)

#### Data are Binomial = list(c(K1, K2, ...Kn), c(N1, N2, ...Nn))

The likelihood function can be expressed as a product of gamma functions divided by another product of gamma functions.

- L(K1|N1,alpha,beta) = ln(Gamma(alpha+beta))+ln(Gamma(alpha+K1))+ln(Gamma(beta+N1-K1))-ln(Gamma(alpha+beta+N))-ln(Gamma(alpha))-ln(Gamma(beta))
- L(data|alpah,beta) = L(K1|N1,alpha,beta)*L(K2|N2,alpha,beta)*...*L(Kn|Nn,alpha,beta)

### Method of Moments

## Related Distributions

- Continuous Uniform - Can be thought of as a special case of Beta
- Binomial - Beta is a conjugate prior

## External Links

Back to Probability_Distributions