BUILT-IN MATHEMATICA SYMBOL

ChiDistribution

ChiDistribution[

]
represents a

distribution with

degrees of freedom.

MORE INFORMATION

ChiDistribution[] represents the distribution for the square root of a random variable. »

ChiDistribution allows to be any positive real number.

ChiDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

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:

Probability density function:

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Cumulative distribution function:

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Mean and variance of a

distribution are related to the Gamma function:

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Median:

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Scope (7)

Generate a set of pseudorandom numbers that are

distributed:

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:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

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 (9)

Parameter influence on the CDF for each

:

ChiDistribution[

] converges to a normal distribution as

->∞:

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

is a not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: