public final class BetaDistribution extends Object
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.
ContinuousDistribution.Sampler
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) . |
public static BetaDistribution of(double alpha, double beta)
alpha
- First shape parameter (must be positive).beta
- Second shape parameter (must be positive).IllegalArgumentException
- if alpha <= 0
or beta <= 0
.public double getAlpha()
public double getBeta()
public double density(double x)
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.
x
- Point at which the PDF is evaluated.x
.public double logDensity(double x)
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.
x
- Point at which the PDF is evaluated.x
.public double cumulativeProbability(double x)
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.x
- Point at which the CDF is evaluated.x
.public double survivalProbability(double x)
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.
x
- Point at which the survival function is evaluated.x
.public double getMean()
For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the mean is:
\[ \frac{\alpha}{\alpha + \beta} \]
public double getVariance()
For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the variance is:
\[ \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \].
public double getSupportLowerBound()
inverseCumulativeProbability(0)
, i.e.
\( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).
The lower bound of the support is always 0.
public double getSupportUpperBound()
inverseCumulativeProbability(1)
, i.e.
\( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).
The upper bound of the support is always 1.
public ContinuousDistribution.Sampler createSampler(UniformRandomProvider rng)
createSampler
in interface ContinuousDistribution
rng
- Generator of uniformly distributed numbers.public double probability(double x0, double x1)
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)
probability
in interface ContinuousDistribution
x0
- Lower bound (exclusive).x1
- Upper bound (inclusive).x0
and x1
, excluding the lower
and including the upper endpoint.public double inverseCumulativeProbability(double p)
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
, orcumulativeProbability(x) - p
.
The bounds may be bracketed for efficiency.inverseCumulativeProbability
in interface ContinuousDistribution
p
- Cumulative probability.p
-quantile of this distribution
(largest 0-quantile for p = 0
).IllegalArgumentException
- if p < 0
or p > 1
public double inverseSurvivalProbability(double p)
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
, orsurvivalProbability(x) - p
.
The bounds may be bracketed for efficiency.inverseSurvivalProbability
in interface ContinuousDistribution
p
- Survival probability.(1-p)
-quantile of this distribution
(largest 0-quantile for p = 1
).IllegalArgumentException
- if p < 0
or p > 1
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