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

Frequency Polygons

Learning Objectives Create and interpret frequency polygons Create and interpret cumulative frequency polygons Create and interpret overlaid frequency polygons Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions . To create a frequency polygon, start just as for histograms , by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the ...
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

Basics of Sampling Techniques

Population                A   population   is a group of individuals(or)aggregate of objects under study.It is also known as universe. The population is divided by (i)finite population  (ii)infinite population, (iii) hypothetical population,  subject to a statistical study . A population includes each element from the set of observations that can be made. (i) Finite population : A population is called finite if it is possible to count its individuals. It may also be called a countable population. The number of vehicles crossing a bridge every day, (ii) Infinite population : Sometimes it is not possible to count the units contained in the population. Such a population is called infinite or uncountable. ex, The number of germs in the body of a patient of malaria is perhaps something which is uncountable   (iii) Hypothetical population : Statistica...
Post a Comment
Read more