BetaDistribution (Apache Commons Statistics Distribution 1.0 API)

java.lang.Object
- org.apache.commons.statistics.distribution.BetaDistribution

All Implemented Interfaces:

ContinuousDistribution
```
public final class BetaDistribution
extends Object
```
Implementation of the beta distribution.
The probability density function of \( X \) is:
\[ f(x; \alpha, \beta) = \frac{1}{ B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1} \]
for \( \alpha > 0 \), \( \beta > 0 \), \( x \in [0, 1] \), and the beta function, \( B \), is a normalization constant:
\[ B(\alpha, \beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} \]
where \( \Gamma \) is the gamma function.
\( \alpha \) and \( \beta \) are shape parameters.

See Also:

Beta distribution (Wikipedia), Beta distribution (MathWorld)

Nested Class Summary
- Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
  ContinuousDistribution.Sampler

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`ContinuousDistribution.Sampler`	`createSampler(UniformRandomProvider rng)` Creates a sampler.
`double`	`cumulativeProbability(double x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
`double`	`density(double x)` Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
`double`	`getAlpha()` Gets the first shape parameter of this distribution.
`double`	`getBeta()` Gets the second shape parameter of this distribution.
`double`	`getMean()` Gets the mean of this distribution.
`double`	`getSupportLowerBound()` Gets the lower bound of the support.
`double`	`getSupportUpperBound()` Gets the upper bound of the support.
`double`	`getVariance()` Gets the variance of this distribution.
`double`	`inverseCumulativeProbability(double p)` Computes the quantile function of this distribution.
`double`	`inverseSurvivalProbability(double p)` Computes the inverse survival probability function of this distribution.
`double`	`logDensity(double x)` Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point `x`.
`static BetaDistribution`	`of(double alpha, double beta)` Creates a beta distribution.
`double`	`probability(double x0, double x1)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(x0 < X <= x1)`.
`double`	`survivalProbability(double x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X > x)`.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Method Detail
  - of
```
public static BetaDistribution of(double alpha,
                                  double beta)
```
    Creates a beta distribution.
    
    Parameters:
    
    alpha - First shape parameter (must be positive).
    
    beta - Second shape parameter (must be positive).
    
    Returns:
    
    the distribution
    
    Throws:
    
    IllegalArgumentException - if alpha <= 0 or beta <= 0.
  - getAlpha
```
public double getAlpha()
```
    Gets the first shape parameter of this distribution.
    
    Returns:
    
    the first shape parameter.
  - getBeta
```
public double getBeta()
```
    Gets the second shape parameter of this distribution.
    
    Returns:
    
    the second shape parameter.
  - density
```
public double density(double x)
```
    Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.
    The density is not defined when x = 0, alpha < 1, or x = 1, beta < 1. In this case the limit of infinity is returned.
    
    Parameters:
    
    x - Point at which the PDF is evaluated.
    
    Returns:
    
    the value of the probability density function at x.
  - logDensity
```
public double logDensity(double x)
```
    Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
    The density is not defined when x = 0, alpha < 1, or x = 1, beta < 1. In this case the limit of infinity is returned.
    
    Parameters:
    
    x - Point at which the PDF is evaluated.
    
    Returns:
    
    the logarithm of the value of the probability density function at x.
  - cumulativeProbability
```
public double cumulativeProbability(double x)
```
    For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.
    
    Parameters:
    
    x - Point at which the CDF is evaluated.
    
    Returns:
    
    the probability that a random variable with this distribution takes a value less than or equal to x.
  - survivalProbability
```
public double survivalProbability(double x)
```
    For a random variable X whose values are distributed according to this distribution, this method returns P(X > x). In other words, this method represents the complementary cumulative distribution function.
    By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.
    
    Parameters:
    
    x - Point at which the survival function is evaluated.
    
    Returns:
    
    the probability that a random variable with this distribution takes a value greater than x.
  - getMean
```
public double getMean()
```
    Gets the mean of this distribution.
    For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the mean is:
    \[ \frac{\alpha}{\alpha + \beta} \]
    
    Returns:
    
    the mean.
  - getVariance
```
public double getVariance()
```
    Gets the variance of this distribution.
    For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the variance is:
    \[ \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \].
    
    Returns:
    
    the variance.
  - getSupportLowerBound
```
public double getSupportLowerBound()
```
    Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).
    The lower bound of the support is always 0.
    
    Returns:
    
    0.
  - getSupportUpperBound
```
public double getSupportUpperBound()
```
    Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).
    The upper bound of the support is always 1.
    
    Returns:
    
    1.
  - createSampler
```
public ContinuousDistribution.Sampler createSampler(UniformRandomProvider rng)
```
    Creates a sampler.
    
    Specified by:
    
    createSampler in interface ContinuousDistribution
    
    Parameters:
    
    rng - Generator of uniformly distributed numbers.
    
    Returns:
    
    a sampler that produces random numbers according this distribution.
  - probability
```
public double probability(double x0,
                          double x1)
```
    For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
    
    Specified by:
    
    probability in interface ContinuousDistribution
    
    Parameters:
    
    x0 - Lower bound (exclusive).
    
    x1 - Upper bound (inclusive).
    
    Returns:
    
    the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.
  - inverseCumulativeProbability
```
public double inverseCumulativeProbability(double p)
```
    Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:
    \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
    The default implementation returns:
    - ContinuousDistribution.getSupportLowerBound() for p = 0,
    - ContinuousDistribution.getSupportUpperBound() for p = 1, or
    - the result of a search for a root between the lower and upper bound using cumulativeProbability(x) - p. The bounds may be bracketed for efficiency.
    Specified by:
    
    inverseCumulativeProbability in interface ContinuousDistribution
    
    Parameters:
    
    p - Cumulative probability.
    
    Returns:
    
    the smallest p-quantile of this distribution (largest 0-quantile for p = 0).
    
    Throws:
    
    IllegalArgumentException - if p < 0 or p > 1
  - inverseSurvivalProbability
```
public double inverseSurvivalProbability(double p)
```
    Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:
    \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
    By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.
    The default implementation returns:
    - ContinuousDistribution.getSupportLowerBound() for p = 1,
    - ContinuousDistribution.getSupportUpperBound() for p = 0, or
    - the result of a search for a root between the lower and upper bound using survivalProbability(x) - p. The bounds may be bracketed for efficiency.
    Specified by:
    
    inverseSurvivalProbability in interface ContinuousDistribution
    
    Parameters:
    
    p - Survival probability.
    
    Returns:
    
    the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).
    
    Throws:
    
    IllegalArgumentException - if p < 0 or p > 1

Class BetaDistribution

Nested Class Summary

Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

Method Summary

Methods inherited from class java.lang.Object

Method Detail

of

getAlpha

getBeta

density

logDensity

cumulativeProbability

survivalProbability

getMean

getVariance

getSupportLowerBound

getSupportUpperBound

createSampler

probability

inverseCumulativeProbability

inverseSurvivalProbability