F Distribution

The F distribution, also known as the variance ratio distribution or the Fisher-Snedecor distribution, is used to model the ratio of two variances. It is primarily used in ANOVA to determine the significance of the variation due to one or more effects compared to the total variation in the sample.

The F distribution has two parameters: the degrees of freedom of the numerator $n$ and the degrees of freedom of the denominator $m$. These parameters act as shape parameters.

The probability density function (PDF) of the F distribution is:

where $n$ is the degrees of freedom of the numerator, and $m$ is the degrees of freedom of the denominator.

The cumulative distribution function (CDF) is:

where $I_x(a,b)$ is the regularized incomplete beta function.

The F distribution is defined for any non-negative real value ($x\ge 0$), with the parameters $n$ and $m$ being strictly positive integers.

The F distribution plays a crucial role in statistical inference and hypothesis testing across multiple fields.

• In Analysis of Variance (ANOVA), the F distribution is used to test whether there are significant differences between the means of multiple groups by comparing the variation between groups to the variation within groups.

• In regression analysis, the F-test helps determine whether a proposed linear model provides a better fit to the data than a model with no predictors.

• The F-test is employed in testing the equality of variances between two populations, which is particularly important in determining whether two samples come from populations with the same variability.

• In multivariate statistics, the F distribution is used in MANOVA (Multivariate Analysis of Variance) to assess the significance of differences between groups across multiple dependent variables simultaneously.

• Quality control applications use the F distribution to monitor process variability and test whether manufacturing processes maintain consistent variance over time.

The F distribution has several notable properties:

• If $X$ follows an F distribution with parameters $n,m$, then $\frac{1}{X}$ follows an F distribution with parameters $m,n$

• The F distribution is related to the ratio of chi-square distributions: if $X\sim F_{n,m}$, then $X\stackrel{d}{=}\frac{U_1/n}{U_2/m}$ where $U_1\sim {\chi }_n^2$ and $U_2\sim {\chi }_m^2$

• The square of a Student's t distribution with $n$ degrees of freedom is equivalent to an F distribution with parameters 1 and $n$

• The F distribution is a ratio of independent chi-square distributions divided by their respective degrees of freedom

The F distribution is implemented by the FDistribution class. It has one constructor which has two parameters. The following constructs an F distribution with 4 degrees of freedom for the numerator, and 25 degrees of freedom for the denominator:

var f = new FDistribution(4, 25);

Dim f = New FDistribution(4, 25)

No code example is currently available or this language may not be supported.

let f = FDistribution(4.0, 25.0)

The FDistribution class has two specific properties, DenominatorDegreesOfFreedom and NumeratorDegreesOfFreedom, which return the parameters of the distribution.

FDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
double sample = FDistribution.Sample(random, 4, 25);

Dim random = New Pcg32()
Dim sample = FDistribution.Sample(random, 4, 25)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = FDistribution.Sample(random, 4.0, 25.0)

The above example uses the Pcg32 class to generate uniform random numbers.

For more information on the F distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 27. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 20. Wiley Series in Probability and Statistics.

• Fisher, R.A. (1925). "Statistical Methods for Research Workers".

• Snedecor, G.W., & Cochran, W.G. (1989). "Statistical Methods".

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