# ChiDistribution

represents a distribution with ν degrees of freedom.

# Details # Background & Context

• represents a statistical distribution parametrized by a positive value ν indicating the degrees of freedom of the distribution. ν determines the general shape of the probability density function (PDF) of a chi distribution and, depending on the values of ν, the PDF may be either monotonic decreasing or may have a single "peak" (i.e. an absolute maximum) with a potential singularity approaching the lower boundary of its domain.
• ChiDistribution is the distribution followed by the square root of a chi-squared random variable. In other words, if is a random variable and (where denotes "is distributed as"), then . The sum of a collection , , , of identically-normally-distributed independent random variables is also chi-distributed. In addition to their statistical significance, chi distributions also arise in a number of scientific applications including the speed distribution of an ideal gas, the accuracy of projectiles under various conditions, and the effects of radar on fluid dispersion.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a chi distribution. Distributed[x,ChiDistribution[ν]], written more concisely as xChiDistribution[ν], can be used to assert that a random variable x is distributed according to a chi distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[ChiDistribution[ν],x] and CDF[ChiDistribution[ν],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a chi distribution, EstimatedDistribution to estimate a chi parametric distribution from given data, and FindDistributionParameters to fit data to a chi distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic chi distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic chi distribution.
• TransformedDistribution can be used to represent a transformed chi distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a chi distribution and ProductDistribution can be used to compute a joint distribution with independent component distributions involving chi distributions.
• ChiDistribution is closely related to a number of other distributions. For example, for and , is equivalent to and , respectively. Moreover, ChiDistribution can be viewed as a special case of the more general NakagamiDistribution in the sense that if XNakagamiDistribution[m,ω] is an independent random variable, then where YChiDistribution[2m]. ChiDistribution can also be obtained as a transformation of NoncentralChiSquareDistribution and is closely related to GammaDistribution, NormalDistribution, and HalfNormalDistribution.

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance of a distribution are related to the Gamma function:

Median:

## Scope(8)

Generate a sample of pseudorandom numbers from a distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness:

In the limit the distribution becomes symmetric:

Kurtosis:

In the limit the distribution has the same kurtosis as NormalDistribution:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Hazard function:

Quantile function:

Use dimensionless Quantity to specify the degree of freedom parameter ν:

## Applications(1)

Components of a 4D vector are normally distributed. Find the distribution of the length of the vector:

Find the average length of the vector:

Simulate possible lengths for a sample of 30 vectors:

## Properties & Relations(8)

converges to a normal distribution as ν->:

Relationships to other distributions: The square of a variable follows the ChiSquareDistribution:

The distribution with is equivalent to HalfNormalDistribution with :

The distribution with is equivalent to RayleighDistribution with :

The distribution with is equivalent to MaxwellDistribution with : distribution is a special case of GammaDistribution:

The norm of standard normally distributed variables is a distribution:

## Possible Issues(2)

ChiDistribution is not defined when ν is a not a positive real number: Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

## Neat Examples(1)

PDFs for different ν values with CDF contours: