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:
$f (x; a, b, c, d) = {\begin{cases} \frac{2}{d + c - a - b} \frac{x - a}{b - a} & for a \leq x < b \\ \frac{2}{d + c - a - b} & for b \leq x < c \\ \frac{2}{d + c - a - b} \frac{d - x}{d - c} & for c \leq x \leq d \end{cases}$
for $- \infty < a \leq b \leq c \leq d < \infty$ 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:
    $\frac{1}{3 (d + c - b - a)} (\frac{d^{3} - c^{3}}{d - c} - \frac{b^{3} - a^{3}}{b - a})$
    
    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:
    $\frac{1}{6 (d + c - b - a)} (\frac{d^{4} - c^{4}}{d - c} - \frac{b^{4} - a^{4}}{b - a}) - μ^{2}$
    where $μ$ 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. $inf {x \in R : P (X \leq x) > 0}$ .
    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. $inf {x \in R : P (X \leq x) = 1}$ .
    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:
    $x = {\begin{cases} inf {x \in R : P (X \leq x) \geq p} & for 0 < p \leq 1 \\ inf {x \in R : P (X \leq x) > 0} & for p = 0 \end{cases}$
    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:
    $x = {\begin{cases} inf {x \in R : P (X > x) \leq p} & for 0 \leq p < 1 \\ inf {x \in R : P (X > x) < 1} & for p = 1 \end{cases}$
    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