by Marco Taboga, PhD
The F distribution is a univariate continuous distribution often used in hypothesis testing.
Table of contents
How it arises
Definition
Relation to the Gamma distribution
Relation to the Chi-square distribution
Expected value
Variance
Higher moments
Moment generating function
Characteristic function
Distribution function
Density plots
Plot 1 - Increasing the first parameter
Plot 2 - Increasing the second parameter
Plot 3 - Increasing both parameters
Solved exercises
Exercise 1
Exercise 2
References
How it arises
A random variable has an F distribution if it can be written as a ratiobetween a Chi-square random variable with degrees of freedom and a Chi-square random variable , independent of , with degrees of freedom (where each variable is divided by its degrees of freedom).
Ratios of this kind occur very often in statistics.
Definition
F random variables are characterized as follows.
Definition Let be a continuous random variable. Let its support be the set of positive real numbers:Let . We say that has an F distribution with and degrees of freedom if and only if its probability density function iswhere is a constant:and is the Beta function.
To better understand the F distribution, you can have a look at its density plots.
Relation to the Gamma distribution
An F random variable can be written as a Gamma random variable with parameters and , where the parameter is equal to the reciprocal of another Gamma random variable, independent of the first one, with parameters and .
Proposition The probability density function of can be written aswhere:
- is the probability density function of a Gamma random variable with parameters and :
- is the probability density function of a Gamma random variable with parameters and :
Proof
We need to prove thatwhereandLet us start from the integrand function: 
where and is the probability density function of a random variable having a Gamma distribution with parameters and . Therefore,
Relation to the Chi-square distribution
In the introduction, we have stated (without a proof) that a random variable has an F distribution with and degrees of freedom if it can be written as a ratiowhere:
- is a Chi-square random variable with degrees of freedom;
- is a Chi-square random variable, independent of , with degrees of freedom.
The statement can be proved as follows.
Proof
This statement is equivalent to the statement proved above (relation to the Gamma distribution): can be thought of as a Gamma random variable with parameters and , where the parameter is equal to the reciprocal of another Gamma random variable , independent of the first one, with parameters and . The equivalence can be proved as follows.
Since a Gamma random variable with parameters and is just the product between the ratio and a Chi-square random variable with degrees of freedom (see the lecture entitled Gamma distribution), we can write where is a Chi-square random variable with degrees of freedom. Now, we know that is equal to the reciprocal of another Gamma random variable , independent of , with parameters and . Therefore,But a Gamma random variable with parameters and is just the product between the ratio and a Chi-square random variable with degrees of freedom. Therefore, we can write
Expected value
The expected value of an F random variable is well-defined only for and it is equal to
Proof
It can be derived thanks to the integral representation of the Beta function:
In the above derivation we have used the properties of the Gamma function and the Beta function. It is also clear that the expected value is well-defined only when : when , the above improper integrals do not converge (both arguments of the Beta function must be strictly positive).
Variance
The variance of an F random variable is well-defined only for and it is equal to
Proof
It can be derived thanks to the usual variance formula () and to the integral representation of the Beta function:
In the above derivation we have used the properties of the Gamma function and the Beta function. It is also clear that the expected value is well-defined only when : when , the above improper integrals do not converge (both arguments of the Beta function must be strictly positive).
Higher moments
The -th moment of an F random variable is well-defined only for and it is equal to
Proof
It is obtained by using the definition of moment:
In the above derivation we have used the properties of the Gamma function and the Beta function. It is also clear that the expected value is well-defined only when : when , the above improper integrals do not converge (both arguments of the Beta function must be strictly positive).
Moment generating function
An F random variable does not possess a moment generating function.
Proof
When a random variable possesses a moment generating function, then the -th moment of exists and is finite for any . But we have proved above that the -th moment of exists only for . Therefore, can not have a moment generating function.
Characteristic function
There is no simple expression for the characteristic function of the F distribution.
It can be expressed in terms of the Confluent hypergeometric function of the second kind (a solution of a certain differential equation, called confluent hypergeometric differential equation).
The interested reader can consult Phillips (1982).
Distribution function
The distribution function of an F random variable iswhere the integralis known as incomplete Beta function and is usually computed numerically with the help of a computer algorithm.
Proof
This is proved as follows:
Density plots
The plots below illustrate how the shape of the density of an F distribution changes when its parameters are changed.
Plot 1 - Increasing the first parameter
The following plot shows two probability density functions (pdfs):
-
the blue line is the pdf of an F random variable with parameters
and ; -
the orange line is the pdf of an F random variable with parameters
and .
By increasing the first parameter from to , the mean of the distribution (vertical line) does not change.
However, part of the density is shifted from the tails to the center of the distribution.
Plot 2 - Increasing the second parameter
In the following plot:
-
the blue line is the density of an F distribution with parameters
and ; -
the orange line is the density of an F distribution with parameters
and .
By increasing the second parameter from to , the mean of the distribution (vertical line) decreases (from to ) and some density is shifted from the tails (mostly from the right tail) to the center of the distribution.
Plot 3 - Increasing both parameters
In the next plot:
-
the blue line is the density of an F random variable with parameters
and ; -
the orange line is the density of an F random variable with parameters
and .
By increasing the two parameters, the mean of the distribution decreases (from to ) and density is shifted from the tails to the center of the distribution. As a result, the distribution has a bell shape similar to the shape of the normal distribution.
Solved exercises
Below you can find some exercises with explained solutions.
Exercise 1
Let be a Gamma random variable with parameters and .
Let be another Gamma random variable, independent of , with parameters and .
Find the expected value of the ratio
Solution
We can writewhere and are two independent Gamma random variables, the parameters of are and and the parameters of are and (see the lecture entitled Gamma distribution). By using this fact, the ratio can be written aswhere has an F distribution with parameters and . Therefore,
Exercise 2
Find the third moment of an F random variable with parameters and .
Solution
We need to use the formula for the -th moment of an F random variable:
Plugging in the parameter values, we obtainwhere we have used the relation between the Gamma function and the factorial function.
References
Phillips, P. C. B. (1982) The true characteristic function of the F distribution, Biometrika, 69, 261-264.
How to cite
Please cite as:
Taboga, Marco (2021). "F distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/F-distribution.
