# Class ChiSquareDistribution

java.lang.Object
smile.stat.distribution.ChiSquareDistribution
All Implemented Interfaces:
`Serializable`, `Distribution`, `ExponentialFamily`

public class ChiSquareDistribution extends Object implements ExponentialFamily
Chi-square (or chi-squared) distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. It's mean and variance are k and 2k, respectively. The chi-square distribution is a special case of the gamma distribution. It follows from the definition of the chi-square distribution that the sum of independent chi-square variables is also chi-square distributed. Specifically, if Xi are independent chi-square variables with ki degrees of freedom, respectively, then Y = Σ Xi is chi-square distributed with Σ ki degrees of freedom.

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.

• ## Field Summary

Fields
Modifier and Type
Field
Description
`final int`
`nu`
The degrees of freedom.
• ## Constructor Summary

Constructors
Constructor
Description
`ChiSquareDistribution(int nu)`
Constructor.
• ## Method Summary

Modifier and Type
Method
Description
`double`
`cdf(double x)`
Cumulative distribution function.
`double`
`entropy()`
Returns Shannon entropy of the distribution.
`int`
`length()`
Returns 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 on the specific distribution.
`double`
`mean()`
Returns 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()`
Returns the standard deviation of distribution.
`String`
`toString()`

`double`
`variance()`
Returns the variance of distribution.

### 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

`inverseTransformSampling, likelihood, logLikelihood, quantile, quantile, rand, rejectionSampling`
• ## Field Details

• ### nu

public final int nu
The degrees of freedom.
• ## Constructor Details

• ### ChiSquareDistribution

public ChiSquareDistribution(int nu)
Constructor.
Parameters:
`nu` - the degree of freedom.
• ## Method Details

• ### length

public int length()
Description copied from interface: `Distribution`
Returns 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`
Returns:
The number of parameters.
• ### mean

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

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

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

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

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

public double rand()
Description copied from interface: `Distribution`
Generates a random number following this distribution.
Specified by:
`rand` in interface `Distribution`
Returns:
a random number.
• ### 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`
Parameters:
`x` - a real number.
Returns:
the density.
• ### 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`
Parameters:
`x` - a real number.
Returns:
the log density.
• ### 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`
Parameters:
`x` - a real number.
Returns:
the probability.
• ### 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`
Parameters:
`p` - the probability.
Returns:
the quantile.
• ### M

public Mixture.Component M(double[] x, double[] posteriori)
Description copied from interface: `ExponentialFamily`
The M step in the EM algorithm, which depends on 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.