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F distribution

A random variable X has an F distribution if it can be written as a ratio:[eq1]between a Chi-square random variable $Y_{1}$ with $n_{1}$ degrees of freedom and a Chi-square random variable $Y_{2}$, independent of $Y_{1}$, with $n_{2}$ 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.

Definition

F random variables are characterized as follows:

Definition Let X be an absolutely continuous random variable. Let its support be the set of positive real numbers:[eq2]Let [eq3]. We say that X has an F distribution with $n_{1}$ and $n_{2}$ degrees of freedom if its probability density function is:[eq4]where $c$ is a constant:[eq5]and $Bleft( {}
ight) $ 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 $n_{1}$ and $h_{1}$, where the parameter $h_{1}$ is equal to the reciprocal of another Gamma random variable, independent of the first one, with parameters $n_{2}$ and $h_{2}=1$:

Proposition (Integral representation) The probability density function of X can be written as:[eq6]where:

  1. [eq7] is the probability density function of a Gamma random variable with parameters $n_{1}$ and $frac{1}{z}$:[eq8]

  2. [eq9] is the probability density function of a Gamma random variable with parameters $n_{2}$ and $h_{2}=1$:[eq10]

Proof

We need to prove that:[eq11]where:[eq12]and[eq13]Let us start from the integrand function: [eq14]where [eq15]and [eq16] is the probability density function of a random variable having a Gamma distribution with parameters $n_{1}+n_{2}$ and [eq17]. Therefore:[eq18]

Relation to the Chi-square distribution

In the introduction, we have stated (without a proof) that a random variable X has an F distribution with $n_{1}$ and $n_{2}$ degrees of freedom if it can be written as a ratio:[eq19]where:

  1. $Y_{1}$ is a Chi-square random variable with $n_{1}$ degrees of freedom

  2. $Y_{2}$ is a Chi-square random variable, independent of $Y_{1}$, with $n_{2}$ 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): X can be thought of as a Gamma random variable with parameters $n_{1}$ and $h_{1}$, where the parameter $h_{1}$ is equal to the reciprocal of another Gamma random variable Z, independent of the first one, with parameters $n_{2}$ and $h_{2}=1$. The equivalence can be proved as follows.

Since a Gamma random variable with parameters $n_{1}$ and $h_{1}$ is just the product between the ratio $h_{1}/n_{1}$ and a Chi-square random variable with $n_{1}$ degrees of freedom (see the lecture entitled Gamma distribution), we can write: [eq20]where $Y_{1}$ is a Chi-square random variable with $n_{1}$ degrees of freedom. Now, we know that $h_{1}$ is equal to the reciprocal of another Gamma random variable Z, independent of $Y_{1}$, with parameters $n_{2}$ and $h_{2}=1$. Therefore:[eq21]But a Gamma random variable with parameters $n_{2}$ and $h_{2}=1$ is just the product between the ratio $1/n_{2}$ and a Chi-square random variable with $n_{2}$ degrees of freedom. Therefore, we can write: [eq22]

Expected value

The expected value of an F random variable X is well-defined only for $n_{2}>2$ and it is equal to:[eq23]

Proof

It can be derived thanks to the integral representation of the Beta function:[eq24]

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 $n_{2}>2$: when $n_{2}leq 2$, 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 X is well-defined only for $n_{2}>4$ and it is equal to:[eq25]

Proof

It can be derived thanks to the usual variance formula ([eq26]) and to the integral representation of the Beta function:[eq27]

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 $n_{2}>4$: when $n_{2}leq 4$, the above improper integrals do not converge (both arguments of the Beta function must be strictly positive).

Higher moments

The k-th moment of an F random variable X is well-defined only for $n_{2}>2k$ and it is equal to:[eq28]

Proof

Using the definition of moment:[eq29]

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 $n_{2}>2k$: when $n_{2}leq 2k$, the above improper integrals do not converge (both arguments of the Beta function must be strictly positive).

Moment generating function

An F random variable X does not possess a moment generating function.

Proof

When a random variable X possesses a moment generating function, then the k-th moment of X exists and is finite for any $kin U{2115} $. But we have proved above that the k-th moment of X exists only for $k<n_{2}/2$. Therefore, X 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 is:[eq30]where the integral[eq31]is known as incomplete Beta function and is usually computed numerically with the help of a computer algorithm.

Proof

This is proved as follows:[eq32]

Solved exercises

Below you can find some exercises with explained solutions:

  1. Exercise set 1

References

Phillips, P. C. B. (1982) The true characteristic function of the F distribution, Biometrika, 69, 261-264.

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