WeibullDistribution

java.lang.Object
- smile.stat.distribution.AbstractDistribution
- - smile.stat.distribution.WeibullDistribution

All Implemented Interfaces:

java.io.Serializable, Distribution
```
public class WeibullDistribution
extends AbstractDistribution
```
The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter. The distribution has two parameters: k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. The probability density function is
```
     f(x;λ,k) = k/λ (x/λ)^k-1e^{-(x/λ)^k}
 
```
for x ≥ 0.
The Weibull distribution is often used in the field of life data analysis due to its flexibility - it can mimic the behavior of other statistical distributions such as the normal and the exponential. If the failure rate decreases over time, then k < 1. If the failure rate is constant over time, then k = 1. If the failure rate increases over time, then k > 1.
An understanding of the failure rate may provide insight as to what is causing the failures:
- A decreasing failure rate would suggest "infant mortality". That is, defective items fail early and the failure rate decreases over time as they fall out of the population.
- A constant failure rate suggests that items are failing from random events.
- An increasing failure rate suggests "wear out" - parts are more likely to fail as time goes on.
Under certain parameterizations, the Weibull distribution reduces to several other familiar distributions:
- When k = 1, it is the exponential distribution.
- When k = 2, it becomes equivalent to the Rayleigh distribution, which models the modulus of a two-dimensional uncorrelated bivariate normal vector.
- When k = 3.4, it appears similar to the normal distribution.
- As k goes to infinity, the Weibull distribution asymptotically approaches the Dirac delta function.
See Also:

Serialized Form

Field Summary

Fields
Modifier and Type Field and Description

double k
The shape parameter.

double lambda
The scale parameter.

Fields
Modifier and Type	Field and Description
`double`	`k` The shape parameter.
`double`	`lambda` The scale parameter.

Constructor Summary

Constructors
Constructor and Description

WeibullDistribution(double k)
Constructor.

WeibullDistribution(double k, double lambda)
Constructor.

Constructors
Constructor and Description
`WeibullDistribution(double k)` Constructor.
`WeibullDistribution(double k, double lambda)` Constructor.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`cdf(double x)` Cumulative distribution function.
`double`	`entropy()` Shannon entropy of the distribution.
`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.
`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.
`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, sd

- Field Detail
  - k
```
public final double k
```
    The shape parameter.
  - lambda
```
public final double lambda
```
    The scale parameter.
- Constructor Detail
  - WeibullDistribution
```
public WeibullDistribution(double k)
```
    Constructor. The default scale parameter is 1.0.
    
    Parameters:
    
    k - the shape parameter.
  - WeibullDistribution
```
public WeibullDistribution(double k,
                           double lambda)
```
    Constructor.
    
    Parameters:
    
    k - the shape parameter.
    
    lambda - the scale parameter.
- Method Detail
  - 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.
  - mean
```
public double mean()
```
    Description copied from interface: Distribution
    
    The mean of distribution.
  - variance
```
public double variance()
```
    Description copied from interface: Distribution
    
    The variance of distribution.
  - entropy
```
public double entropy()
```
    Description copied from interface: Distribution
    
    Shannon entropy of the 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.
  - 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.
  - logp
```
public double logp(double x)
```
    Description copied from interface: Distribution
    
    The density at x in log scale, which may prevents the underflow problem.
  - cdf
```
public double cdf(double x)
```
    Description copied from interface: Distribution
    
    Cumulative distribution function. That is the probability to the left of x.
  - 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.

Class WeibullDistribution

Field Summary

Constructor Summary

Method Summary

Methods inherited from class smile.stat.distribution.AbstractDistribution

Methods inherited from class java.lang.Object

Methods inherited from interface smile.stat.distribution.Distribution

Field Detail

k

lambda

Constructor Detail

WeibullDistribution

WeibullDistribution

Method Detail

length

mean

variance

entropy

toString

rand

p

logp

cdf

quantile