dev.nm.stat.distribution.univariate

Class ExponentialDistribution

java.lang.Object

dev.nm.stat.distribution.univariate.ExponentialDistribution

All Implemented Interfaces:

ProbabilityDistribution

public class ExponentialDistribution
extends Object
implements ProbabilityDistribution

The exponential distribution describes the times between events in a Poisson process, a process in which events occur continuously and independently at a constant average rate.

The R equivalent functions are dexp, pexp, qexp, rexp.

Waiting-time probabilities

Supply a strictly positive rate per unit of time. A rate of two per minute means an average wait of half a minute; quantiles use the same time unit.

 import dev.nm.stat.distribution.univariate.ExponentialDistribution;
 ExponentialDistribution waiting = new ExponentialDistribution(2.0);
 assert waiting.mean() == 0.5;
 double medianWait = waiting.quantile(0.5);
 assert Math.abs(medianWait - Math.log(2) / 2) < 1e-12;
 assert Math.abs(waiting.cdf(medianWait) - 0.5) < 1e-12;
 

See Also:

Wikipedia: ExponentialDistribution distribution

Constructor Summary

Constructors

Constructor and Description
ExponentialDistribution() Construct an instance of the standard exponential distribution, where the rate/lambda is 1.
ExponentialDistribution(double lambda) Construct an exponential distribution.

Method Summary

Modifier and Type	Method and Description
double	cdf(double x) Gets the cumulative probability F(x) = Pr(X ≤ x).
double	density(double x) The density function, which, if exists, is the derivative of F.
double	entropy() Gets the entropy of this distribution.
double	kurtosis() Gets the excess kurtosis of this distribution.
double	mean() Gets the mean of this distribution.
double	median() Gets the median of this distribution.
double	moment(double t) The moment generating function is the expected value of \(e^{tX}\).
double	quantile(double u) Gets the quantile, the inverse of the cumulative distribution function.
double	skew() Gets the skewness of this distribution.
double	variance() Gets the variance of this distribution.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

ExponentialDistribution

public ExponentialDistribution()

Construct an instance of the standard exponential distribution, where the rate/lambda is 1.

ExponentialDistribution

public ExponentialDistribution(double lambda)

Construct an exponential distribution.

Parameters:

lambda - the rate

Method Detail

mean

public double mean()

Description copied from interface: ProbabilityDistribution

Gets the mean of this distribution.

Specified by:

mean in interface ProbabilityDistribution

Returns:

the mean

See Also:

Wikipedia: Expected value

median

public double median()

Description copied from interface: ProbabilityDistribution

Gets the median of this distribution.

Specified by:

median in interface ProbabilityDistribution

Returns:

the median

See Also:

Wikipedia: Median

variance

public double variance()

Description copied from interface: ProbabilityDistribution

Gets the variance of this distribution.

Specified by:

variance in interface ProbabilityDistribution

Returns:

the variance

See Also:

Wikipedia: Variance

skew

public double skew()

Description copied from interface: ProbabilityDistribution

Gets the skewness of this distribution.

Specified by:

skew in interface ProbabilityDistribution

Returns:

the skewness

See Also:

Wikipedia: Skewness

kurtosis

public double kurtosis()

Description copied from interface: ProbabilityDistribution

Gets the excess kurtosis of this distribution.

Specified by:

kurtosis in interface ProbabilityDistribution

Returns:

the excess kurtosis

See Also:

Wikipedia: Kurtosis

entropy

public double entropy()

Description copied from interface: ProbabilityDistribution

Gets the entropy of this distribution.

Specified by:

entropy in interface ProbabilityDistribution

Returns:

the entropy

See Also:

Wikipedia: Entropy (information theory)

cdf

public double cdf(double x)

Description copied from interface: ProbabilityDistribution

Gets the cumulative probability F(x) = Pr(X ≤ x).

Specified by:

cdf in interface ProbabilityDistribution

Parameters:

x - x

Returns:

F(x) = Pr(X ≤ x)

See Also:

Wikipedia: Cumulative distribution function

quantile

public double quantile(double u)

Description copied from interface: ProbabilityDistribution

Gets the quantile, the inverse of the cumulative distribution function. It is the value below which random draws from the distribution would fall u×100 percent of the time. \[ \begin{gathered} F^{-1}(u) = x, such that \\ Pr(X \le x) = u \end{gathered} \] This may not always exist.

Specified by:

quantile in interface ProbabilityDistribution

Parameters:

u - u, a quantile

Returns:

\(F^{-1}(u)\)

See Also:

Wikipedia: Quantile function

density

public double density(double x)

Description copied from interface: ProbabilityDistribution

The density function, which, if exists, is the derivative of F. It describes the density of probability at each point in the sample space. \[ f(x) = dF(X) / dx \] This may not always exist.

For the discrete cases, this is the probability mass function. It gives the probability that a discrete random variable is exactly equal to some value.

Specified by:

density in interface ProbabilityDistribution

Parameters:

x - x

Returns:

f(x)

See Also:

• Wikipedia: Probability density function
• Wikipedia: Probability mass function

moment

public double moment(double t)

Description copied from interface: ProbabilityDistribution

The moment generating function is the expected value of \(e^{tX}\). That is, \[ E(e^{tX}) \] This may not always exist.

Specified by:

moment in interface ProbabilityDistribution

Parameters:

t - t

Returns:

E(exp(tX))

See Also:

Wikipedia: Moment-generating function