BUILT-IN MATHEMATICA SYMBOL

FRatioDistribution

represents an F-ratio distribution with n numerator and m denominator degrees of freedom.

MORE INFORMATION

FRatioDistribution is also known as the Fisher-Snedecor distribution.

The probability density for value in an F-ratio distribution is proportional to for , and is zero for . »

For integer n and m, the F-ratio distribution gives the distribution of the ratio of variances for samples from normal distributions.

FRatioDistribution allows n and m to be any positive real numbers.

FRatioDistribution 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:

Median:

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Median:

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Scope (7)

Generate a set of pseudorandom numbers that have the F-ratio 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 varies with the degrees of freedom:

Limiting value:

Kurtosis varies with the degrees of freedom:

Limiting value is the kurtosis of 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)

FRatioDistribution is the distribution of the ratio of two sample variances drawn from two normal distributions. Define the Fisher ratio statistic:

Generate 1000 batches of samples from two standard normal distributions:

Compute values of the Fisher ratio statistics for each batch:

Find the

-value of the Fisher ratio test for the first batch:

Compare with FisherRatioTest:

Properties & Relations (12)

Parameter influence on the CDF for each

:

Relationships to other distributions:

ChiSquareDistribution is a limiting case of F-ratio distribution:

F-ratio is the ratio of two ChiSquareDistribution variables:

F-ratio distribution can be obtained from BetaDistribution:

A square of StudentTDistribution has F-ratio distribution:

F-ratio distribution is the distribution of the inverse square of StudentTDistribution:

F-ratio distribution is a special case of type 6 PearsonDistribution:

FRatioDistribution is a transformation of Laplace distribution:

FisherZDistribution is a transformation of F-ratio distribution:

NoncentralFRatioDistribution simplifies to F-ratio distribution:

Doubly NoncentralFRatioDistribution simplifies to F-ratio distribution:

Possible Issues (2)

FRatioDistribution is not defined when n or m is not a positive real number:

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