Skip to main content

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

Comments

Post a Comment

Popular posts from this blog

Runs Test for Detecting Non-randomness

Runs Test for Detecting Non-randomness Purpose: Detect Non-Randomness The runs test ( Bradley, 1968 ) can be used to decide if a data set is from a random process. A run is defined as a series of increasing values or a series of decreasing values. The number of increasing, or decreasing, values is the length of the run. In a random data set, the probability that the ( I +1)th value is larger or smaller than the I th value follows a binomial distribution , which forms the basis of the runs test. Typical Analysis and Test Statistics The first step in the runs test is to count the number of runs in the data sequence. There are several ways to define runs in the literature, however, in all cases the formulation must produce a dichotomous sequence of values. For example, a series of 20 coin tosses might produce the f...
Post a Comment
Read more

Lognormal distribution

Lognormal Distribution Probability Density Function A variable X is lognormally distributed if Y = LN(X) is normally distributed with "LN" denoting the natural logarithm. The general formula for the  probability density function  of the lognormal distribution is where   is the  shape parameter ,   is the  location parameter  and  m is the  scale parameter . The case where   = 0 and  m  = 1 is called the  standard lognormal distribution . The case where   equals zero is called the 2-parameter lognormal distribution. The equation for the standard lognormal distribution is Since the general form of probability functions can be  expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the lognormal probability density function for four values of  . There are several commo...
Post a Comment
Read more

Double exponential distribution

Double Exponential Distribution Probability Density Function The general formula for the  probability density function  of the double exponential distribution is where   is the  location parameter  and   is the  scale parameter . The case where   = 0 and   = 1 is called the  standard double exponential distribution . The equation for the standard double exponential distribution is Since the general form of probability functions can be  expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the double exponential probability density function. Cumulative Distribution Function The formula for the  cumulative distribution function  of the double exponential distribution is The following is the plot of the double exponential cumulative distribution function. Percent Point Function...
Post a Comment
Read more