org.apache.commons.statistics.distribution

Class GeometricDistribution

java.lang.Object

org.apache.commons.statistics.distribution.GeometricDistribution

All Implemented Interfaces:

DiscreteDistribution

public final class GeometricDistribution
extends Object

Implementation of the geometric distribution.

The probability mass function of $X$ is:

$$f(k;p)=(1-p{)}^{k}p$$

for $p\in (0,1]$ the probability of success and $k\in \{0,1,2,\dots \}$ the number of failures.

This parameterization is used to model the number of failures until the first success.

See Also:

Geometric distribution (Wikipedia), Geometric distribution (MathWorld)

Nested Class Summary

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

DiscreteDistribution.Sampler

Method Summary

Modifier and Type	Method and Description
DiscreteDistribution.Sampler	createSampler(UniformRandomProvider rng) Creates a sampler.
double	cumulativeProbability(int x) For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	getMean() Gets the mean of this distribution.
double	getProbabilityOfSuccess() Gets the probability of success parameter of this distribution.
int	getSupportLowerBound() Gets the lower bound of the support.
int	getSupportUpperBound() Gets the upper bound of the support.
double	getVariance() Gets the variance of this distribution.
int	inverseCumulativeProbability(double p) Computes the quantile function of this distribution.
int	inverseSurvivalProbability(double p) Computes the inverse survival probability function of this distribution.
double	logProbability(int x) For a random variable X whose values are distributed according to this distribution, this method returns log(P(X = x)), where log is the natural logarithm.
static GeometricDistribution	of(double p) Creates a geometric distribution.
double	probability(int x) For a random variable X whose values are distributed according to this distribution, this method returns P(X = x).
double	probability(int x0, int x1) For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).
double	survivalProbability(int 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 GeometricDistribution of(double p)

Creates a geometric distribution.

Parameters:

p - Probability of success.

Returns:

the geometric distribution

Throws:

IllegalArgumentException - if p <= 0 or p > 1.

getProbabilityOfSuccess

public double getProbabilityOfSuccess()

Gets the probability of success parameter of this distribution.

Returns:

the probability of success.

probability

public double probability(int 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 probability mass function (PMF) for the distribution.

Parameters:

x - Point at which the PMF is evaluated.

Returns:

the value of the probability mass function at x.

logProbability

public double logProbability(int x)

For a random variable X whose values are distributed according to this distribution, this method returns log(P(X = x)), where log is the natural logarithm.

Parameters:

x - Point at which the PMF is evaluated.

Returns:

the logarithm of the value of the probability mass function at x.

cumulativeProbability

public double cumulativeProbability(int 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(int 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.

inverseCumulativeProbability

public int inverseCumulativeProbability(double p)

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{Z}:P(X\le x)\ge p\}& \text{for}0<p\le 1\\ inf\{x\in \mathbb{Z}:P(X\le x)>0\}& \text{for}p=0\end{array}$$

If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. In this case the result of cumulativeProbability(x) called using the returned p-quantile may not compute the original p.

The default implementation returns:

• DiscreteDistribution.getSupportLowerBound() for p = 0,
• DiscreteDistribution.getSupportUpperBound() for p = 1, or
• the result of a binary search between the lower and upper bound using cumulativeProbability(x). The bounds may be bracketed for efficiency.

Specified by:

inverseCumulativeProbability in interface DiscreteDistribution

Parameters:

p - Cumulative probability.

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

inverseSurvivalProbability

public int 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:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{Z}:P(X\ge x)\le p\}& \text{for}0\le p<1\\ inf\{x\in \mathbb{Z}:P(X\ge x)<1\}& \text{for}p=1\end{array}$$

If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. In this case the result of survivalProbability(x) called using the returned (1-p)-quantile may not compute the original p.

By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

The default implementation returns:

• DiscreteDistribution.getSupportLowerBound() for p = 1,
• DiscreteDistribution.getSupportUpperBound() for p = 0, or
• the result of a binary search between the lower and upper bound using survivalProbability(x). The bounds may be bracketed for efficiency.

Specified by:

inverseSurvivalProbability in interface DiscreteDistribution

Parameters:

p - Cumulative probability.

Returns:

the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

getMean

public double getMean()

Gets the mean of this distribution.

For probability parameter $p$, the mean is:

$$\frac{1-p}{p}$$

Returns:

the mean.

getVariance

public double getVariance()

Gets the variance of this distribution.

For probability parameter $p$, the variance is:

$$\frac{1-p}{p^2}$$

Returns:

the variance.

getSupportLowerBound

public int getSupportLowerBound()

Gets the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0), i.e. $inf\{x\in \mathbb{Z}:P(X\le x)>0\}$. By convention, Integer.MIN_VALUE should be substituted for negative infinity.

The lower bound of the support is always 0.

Returns:

0.

getSupportUpperBound

public int getSupportUpperBound()

Gets the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1), i.e. $inf\{x\in \mathbb{Z}:P(X\le x)=1\}$. By convention, Integer.MAX_VALUE should be substituted for positive infinity.

The upper bound of the support is positive infinity except for the probability parameter p = 1.0.

Returns:

Integer.MAX_VALUE or 0.

createSampler

public DiscreteDistribution.Sampler createSampler(UniformRandomProvider rng)

Creates a sampler.

Specified by:

createSampler in interface DiscreteDistribution

Parameters:

rng - Generator of uniformly distributed numbers.

Returns:

a sampler that produces random numbers according this distribution.

probability

public double probability(int x0,
                          int 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)

Special cases:

• returns 0.0 if x0 == x1;
• returns probability(x1) if x0 + 1 == x1;

Specified by:

probability in interface DiscreteDistribution

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.