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F distribution

F Distribution

Probability Density FunctionThe F distribution is the ratio of two chi-square distributions with degrees of freedom nu1 and nu2, respectively, where each chi-square has first been divided by its degrees of freedom. The formula for the probability density function of the F distribution is
    [GAMMA((nu1+nu2)/2)*(nu1/nu2)**(nu1/2)*x**((nu2/2)-1)]/ GAMMA(nu1/2)*GAMMA(nu2/2)*(1 + Nu1*x/Nu2)**((nu1+nu2)/2)]
where nu1 and nu2 are the shape parameters and  is the gamma function. The formula for the gamma function is
    GAMMA(a) = INTEGRAL[t**(a-1)*EXP(-t)dt] where the integration is from 0 to infinity
In a testing context, the F distribution is treated as a "standardized distribution" (i.e., no location or scale parameters). However, in a distributional modeling context (as with other probability distributions), the F distribution itself can be transformed with alocation parameter, mu, and a scale parameter, sigma.The following is the plot of the F probability density function for 4 different values of the shape parameters.
plot of the F probability density function for 4 different values of the shape parameters
Cumulative Distribution FunctionThe formula for the Cumulative distribution function of the F distribution is
    F(x) = 1 - I(k)(nu2/2,nu1/2)
where k = nu2/ (nu2 + nu1*x) and Ik is the incomplete beta function. The formula for the incomplete beta function is
    I(k)(x,alpha,beta) = INTEGRAL[t**(alpha-1)*(1-t)**(beta-1)dt]/ B(alpha,beta) where the integration is from 0 to x
where B is the beta function
    B(alpha,beta) = INTEGRAL[t**(alpha-1)*(1-t)**(beta-1)dt where the integration is from 0 to 1
The following is the plot of the F cumulative distribution function with the same values of nu1 and nu2 as the pdf plots above.plot of the F cumulative distribution function with the same values of nu1 and nu2 as the pdf plots above
Percent Point FunctionThe formula for the percent point function of the F distribution does not exist in a simple closed form. It is computed numerically.The following is the plot of the F percent point function with the same values of nu1 and nu2 as the pdf plots above.
plot of the F percent point function with the same values of nu1 and nu2 as the pdf plots above
Other Probability FunctionsSince the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
Common StatisticsThe formulas below are for the case where the location parameter is zero and the scale parameter is one.
Meannu2/(n2-2) nu2 > 2
Modenu2*(nu1-2)/(nu1*(nu2+2)) nu1 > 2
Range0 to positive infinity
Standard DeviationSQRT(2*nu2**2*(nu1+nu2-2)/(nu1*(nu2-2)**2*(nu2-4))) nu2 > 4
Coefficient of VariationSQRT(2*(nu1+nu2-2)/(nu1*(nu2-4))) nu2 > 4
Skewness(2*nu1+nu2-2)*SQRT(8*(nu2-4))/(SQRT(nu1)*(nu2-6)*SQRT(nu1+nu2-2)) nu2 > 6
Parameter EstimationSince the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.
CommentsThe F distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the analysis of variance and the F test to determine if the variances of two populations are equal.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the F distribution

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