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

All Implemented Interfaces:

ContinuousDistribution
```
public abstract class TrapezoidalDistribution
extends Object
```
Implementation of the trapezoidal distribution.
The probability density function of \( X \) is:
\[ f(x; a, b, c, d) = \begin{cases} \frac{2}{d+c-a-b}\frac{x-a}{b-a} & \text{for } a\le x \lt b \\ \frac{2}{d+c-a-b} & \text{for } b\le x \lt c \\ \frac{2}{d+c-a-b}\frac{d-x}{d-c} & \text{for } c\le x \le d \end{cases} \]
for \( -\infty \lt a \le b \le c \le d \lt \infty \) and \( x \in [a, d] \).
Note the special cases:
- \( b = c \) is the triangular distribution
- \( a = b \) and \( c = d \) is the uniform distribution
See Also:

Trapezoidal distribution (Wikipedia)

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

Field Summary

Fields
Modifier and Type	Field	Description
`protected double`	`a`	Lower limit of this distribution (inclusive).
`protected double`	`b`	Start of the trapezoid constant density.
`protected double`	`c`	End of the trapezoid constant density.
`protected double`	`d`	Upper limit of this distribution (inclusive).

Method Summary

All Methods Static Methods Instance Methods Abstract Methods Concrete Methods
Modifier and Type	Method	Description
`ContinuousDistribution.Sampler`	`createSampler(org.apache.commons.rng.UniformRandomProvider rng)`	Creates a sampler.
`double`	`getB()`	Gets the start of the constant region of the density function.
`double`	`getC()`	Gets the end of the constant region of the density function.
`abstract 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.
`abstract 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.
`static TrapezoidalDistribution`	`of(double a, double b, double c, double d)`	Creates a trapezoidal 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)`.

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

Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
cumulativeProbability, density, logDensity, survivalProbability

- Field Detail
  - a
```
protected final double a
```
    Lower limit of this distribution (inclusive).
  - b
```
protected final double b
```
    Start of the trapezoid constant density.
  - c
```
protected final double c
```
    End of the trapezoid constant density.
  - d
```
protected final double d
```
    Upper limit of this distribution (inclusive).
- Method Detail
  - of
```
public static TrapezoidalDistribution of(double a,
                                         double b,
                                         double c,
                                         double d)
```
    Creates a trapezoidal distribution.
    The distribution density is represented as an up sloping line from a to b, constant from b to c, and then a down sloping line from c to d.
    
    Parameters:
    
    a - Lower limit of this distribution (inclusive).
    
    b - Start of the trapezoid constant density (first shape parameter).
    
    c - End of the trapezoid constant density (second shape parameter).
    
    d - Upper limit of this distribution (inclusive).
    
    Returns:
    
    the distribution
    
    Throws:
    
    IllegalArgumentException - if a >= d, if b < a, if c < b or if c > d.
  - getMean
```
public abstract double getMean()
```
    Gets the mean of this distribution.
    For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the mean is:
    \[ \frac{1}{3(d+c-b-a)}\left(\frac{d^3-c^3}{d-c}-\frac{b^3-a^3}{b-a}\right) \]
    
    Returns:
    
    the mean.
  - getVariance
```
public abstract double getVariance()
```
    Gets the variance of this distribution.
    For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the variance is:
    \[ \frac{1}{6(d+c-b-a)}\left(\frac{d^4-c^4}{d-c}-\frac{b^4-a^4}{b-a}\right) - \mu^2 \]
    where \( \mu \) is the mean.
    
    Returns:
    
    the variance.
  - getB
```
public double getB()
```
    Gets the start of the constant region of the density function.
    This is the first shape parameter b of the distribution.
    
    Returns:
    
    the first shape parameter b
  - getC
```
public double getC()
```
    Gets the end of the constant region of the density function.
    This is the second shape parameter c of the distribution.
    
    Returns:
    
    the second shape parameter c
  - 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 equal to the lower limit parameter a of the distribution.
    
    Returns:
    
    the lower bound of the support.
  - 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 equal to the upper limit parameter d of the distribution.
    
    Returns:
    
    the upper bound of the support.
  - 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:
    \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
    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
  - createSampler
```
public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.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.

Class TrapezoidalDistribution

Nested Class Summary

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

Field Summary

Method Summary

Methods inherited from class java.lang.Object

Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

Field Detail

a

b

c

d

Method Detail

of

getMean

getVariance

getB

getC

getSupportLowerBound

getSupportUpperBound

probability

inverseCumulativeProbability

inverseSurvivalProbability

createSampler