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 of the two random variables has been divided by its degrees of freedom). The importance of the F distribution stems from the fact that ratios of this kind are encountered very often in statistics.
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F random variables are characterized as follows.
Definition Let be an absolutely 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 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.
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 (Integral representation) 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 :
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,
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.
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
The expected value of an F random variable is well-defined only for and it is equal to
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).
The variance of an F random variable is well-defined only for and it is equal to
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).
The -th moment of an F random variable is well-defined only for and it is equal to
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).
An F random variable does not possess a moment generating function.
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.
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).
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.
This is proved as follows:
The plots below illustrate how the shape of the density of an F distribution changes when its parameters are changed.
The following plot contains the graphs of two F probability density functions:
the first graph (blue line) is the probability density function of an F random variable with parameters and ;
the second graph (red line) is the probability density function of an F random variable with parameters and .
By increasing the first parameter from to , the mean of the distribution (represented by the vertical line in the graph) does not change, however density is shifted from the tails to the center of the distribution.
In the following plot:
the first graph (blue line) is the density of an F distribution with parameters and ;
the second graph (red line) is the density of an F distribution with parameters and .
By increasing the second parameter from to , the mean of the distribution (represented by the vertical line in the graph) decreases (from to ) and density is shifted from the tails (mostly from the right tail) to the center of the distribution.
In the next plot:
the first graph (blue line) is the probability density function of an F random variable with parameters and ;
the second graph (red line) is the probability density function 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 takes a bell shape similar to the shape of the normal distribution.
Below you can find some exercises with explained solutions.
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
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,
Find the third moment of an F random variable with parameters and .
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.
Phillips, P. C. B. (1982) The true characteristic function of the F distribution, Biometrika, 69, 261-264.
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