smile.stat.distribution

## Class ExponentialDistribution

• All Implemented Interfaces:
java.io.Serializable, Distribution, ExponentialFamily

```public class ExponentialDistribution
extends AbstractDistribution
implements ExponentialFamily```
An exponential distribution describes the times between events in a Poisson process, in which events occur continuously and independently at a constant average rate. Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between mutations on a DNA strand. In real world scenarios, the assumption of a constant rate is rarely satisfied. But if we focus on a time interval during which the rate is roughly constant, the exponential distribution can be used as a good approximate model.

The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.

The probability density function of an exponential distribution is f(x; λ) = λe-λx for x ≥ 0. The cumulative distribution function is given by F(x; λ) = 1 - e-λ x for x ≥ 0. An important property of the exponential distribution is that it is memoryless. This means that if a random variable T is exponentially distributed, its conditional probability obeys Pr(T > s + t | T > s) = Pr(T > t) for all s, t ≥ 0.

In queuing theory, the service times of agents in a system are often modeled as exponentially distributed variables. Reliability theory and reliability engineering also make extensive use of the exponential distribution. Because of the memoryless property of this distribution, it is well-suited to model the constant hazard rate portion of the bathtub curve used in reliability theory. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the "failure rates" here are not constant: more failures occur for very young and for very old systems.

Serialized Form
• ### Field Summary

Fields
Modifier and Type Field and Description
`double` `lambda`
The rate parameter.
• ### Constructor Summary

Constructors
Constructor and Description
`ExponentialDistribution(double lambda)`
Constructor.
• ### Method Summary

All Methods
Modifier and Type Method and Description
`double` `cdf(double x)`
Cumulative distribution function.
`double` `entropy()`
Shannon entropy of the distribution.
`static ExponentialDistribution` `fit(double[] data)`
Estimates the distribution parameters by MLE.
`int` `length()`
The number of parameters of the distribution.
`double` `logp(double x)`
The density at x in log scale, which may prevents the underflow problem.
`Mixture.Component` ```M(double[] x, double[] posteriori)```
The M step in the EM algorithm, which depends the specific distribution.
`double` `mean()`
The mean of distribution.
`double` `p(double x)`
The probability density function for continuous distribution or probability mass function for discrete distribution at x.
`double` `quantile(double p)`
The quantile, the probability to the left of quantile is p.
`double` `rand()`
Generates a random number following this distribution.
`double` `sd()`
The standard deviation of distribution.
`java.lang.String` `toString()`
`double` `variance()`
The variance of distribution.
• ### Methods inherited from class smile.stat.distribution.AbstractDistribution

`inverseTransformSampling, quantile, quantile, rejection`
• ### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait`
• ### Methods inherited from interface smile.stat.distribution.Distribution

`likelihood, logLikelihood, rand`
• ### Field Detail

• #### lambda

`public final double lambda`
The rate parameter.
• ### Constructor Detail

• #### ExponentialDistribution

`public ExponentialDistribution(double lambda)`
Constructor.
Parameters:
`lambda` - rate parameter.
• ### Method Detail

• #### fit

`public static ExponentialDistribution fit(double[] data)`
Estimates the distribution parameters by MLE.
• #### length

`public int length()`
Description copied from interface: `Distribution`
The number of parameters of the distribution. The "length" is in the sense of the minimum description length principle.
Specified by:
`length` in interface `Distribution`
• #### mean

`public double mean()`
Description copied from interface: `Distribution`
The mean of distribution.
Specified by:
`mean` in interface `Distribution`
• #### variance

`public double variance()`
Description copied from interface: `Distribution`
The variance of distribution.
Specified by:
`variance` in interface `Distribution`
• #### sd

`public double sd()`
Description copied from interface: `Distribution`
The standard deviation of distribution.
Specified by:
`sd` in interface `Distribution`
• #### entropy

`public double entropy()`
Description copied from interface: `Distribution`
Shannon entropy of the distribution.
Specified by:
`entropy` in interface `Distribution`
• #### toString

`public java.lang.String toString()`
Overrides:
`toString` in class `java.lang.Object`
• #### rand

`public double rand()`
Description copied from interface: `Distribution`
Generates a random number following this distribution.
Specified by:
`rand` in interface `Distribution`
• #### p

`public double p(double x)`
Description copied from interface: `Distribution`
The probability density function for continuous distribution or probability mass function for discrete distribution at x.
Specified by:
`p` in interface `Distribution`
• #### logp

`public double logp(double x)`
Description copied from interface: `Distribution`
The density at x in log scale, which may prevents the underflow problem.
Specified by:
`logp` in interface `Distribution`
• #### cdf

`public double cdf(double x)`
Description copied from interface: `Distribution`
Cumulative distribution function. That is the probability to the left of x.
Specified by:
`cdf` in interface `Distribution`
• #### quantile

`public double quantile(double p)`
Description copied from interface: `Distribution`
The quantile, the probability to the left of quantile is p. It is actually the inverse of cdf.
Specified by:
`quantile` in interface `Distribution`
• #### M

```public Mixture.Component M(double[] x,
double[] posteriori)```
Description copied from interface: `ExponentialFamily`
The M step in the EM algorithm, which depends the specific distribution.
Specified by:
`M` in interface `ExponentialFamily`
Parameters:
`x` - the input data for estimation
`posteriori` - the posteriori probability.
Returns:
the (unnormalized) weight of this distribution in the mixture.