# Noncentral F-distribution

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In probability theory and statistics, the **noncentral F-distribution** is a continuous probability distribution that is a generalization of the (ordinary) F-distribution. It describes the distribution of the quotient (*X*/*n*_{1})/(*Y*/*n*_{2}), where the numerator *X* has a noncentral chi-squared distribution with *n*_{1} degrees of freedom and the denominator *Y* has a central chi-squared distribution *n*_{2} degrees of freedom. It is also required that *X* and *Y* are statistically independent of each other.

It is the distribution of the test statistic in analysis of variance problems when the null hypothesis is false. The noncentral F-distribution is used to find the power function of such a test.

## Occurrence and specification Edit

If is a noncentral chi-squared random variable with noncentrality parameter and degrees of freedom, and is a chi-squared random variable with degrees of freedom that is statistically independent of , then

is a noncentral F-distributed random variable.
The probability density function for the noncentral F-distribution is
^{[1]}

when and zero otherwise. The degrees of freedom and are positive. The noncentrality parameter is nonnegative. The term is the beta function, where

The cumulative distribution function for the noncentral F-distribution is

where is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

and

## Special cases Edit

When *λ* = 0, the noncentral F-distribution becomes the
F-distribution.

## Related distributions Edit

*Z* has a noncentral chi-squared distribution if

where *F* has a noncentral F-distribution.

## Implementations Edit

The noncentral F-distribution is implemented in the R language (e.g., pf function), in MATLAB (ncfcdf, ncfinv, ncfpdf, ncfrnd and ncfstat functions in the statistics toolbox) in Mathematica (NoncentralFRatioDistribution function), in NumPy (random.noncentral_f), and in Boost C++ Libraries.^{[2]}

A collaborative wiki page implements an interactive online calculator, programmed in R language, for noncentral t, chisquare, and F, at the Institute of Statistics and Econometrics, School of Business and Economics, Humboldt-Universität zu Berlin.^{[3]}

## Notes Edit

- ↑ S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p. 29.
- ↑ John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde. Noncentral F Distribution: Boost 1.39.0.
*Boost.org*. URL accessed on 20 August 2011. - ↑ Sigbert Klinke. Comparison of noncentral and central distributions. Humboldt-Universität zu Berlin.

## References Edit

- Weisstein, Eric W., et al Noncentral
*F*-distribution.*MathWorld*. Wolfram Research, Inc. URL accessed on 20 August 2011.

## External links Edit

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