F distribution | Properties, proofs, exercises (2024)

by Marco Taboga, PhD

The F distribution is a univariate continuous distribution often used in hypothesis testing.

F distribution | Properties, proofs, exercises (1)

Table of contents

  1. How it arises

  2. Definition

  3. Relation to the Gamma distribution

  4. Relation to the Chi-square distribution

  5. Expected value

  6. Variance

  7. Higher moments

  8. Moment generating function

  9. Characteristic function

  10. Distribution function

  11. Density plots

    1. Plot 1 - Increasing the first parameter

    2. Plot 2 - Increasing the second parameter

    3. Plot 3 - Increasing both parameters

  12. Solved exercises

    1. Exercise 1

    2. Exercise 2

  13. References

How it arises

A random variable F distribution | Properties, proofs, exercises (2) has an F distribution if it can be written as a ratioF distribution | Properties, proofs, exercises (3)between a Chi-square random variable F distribution | Properties, proofs, exercises (4) with F distribution | Properties, proofs, exercises (5) degrees of freedom and a Chi-square random variable F distribution | Properties, proofs, exercises (6), independent of F distribution | Properties, proofs, exercises (7), with F distribution | Properties, proofs, exercises (8) 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 F distribution | Properties, proofs, exercises (9) be a continuous random variable. Let its support be the set of positive real numbers:F distribution | Properties, proofs, exercises (10)Let F distribution | Properties, proofs, exercises (11). We say that F distribution | Properties, proofs, exercises (12) has an F distribution with F distribution | Properties, proofs, exercises (13) and F distribution | Properties, proofs, exercises (14) degrees of freedom if and only if its probability density function isF distribution | Properties, proofs, exercises (15)where F distribution | Properties, proofs, exercises (16) is a constant:F distribution | Properties, proofs, exercises (17)and F distribution | Properties, proofs, exercises (18) 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 F distribution | Properties, proofs, exercises (19) and F distribution | Properties, proofs, exercises (20), where the parameter F distribution | Properties, proofs, exercises (21) is equal to the reciprocal of another Gamma random variable, independent of the first one, with parameters F distribution | Properties, proofs, exercises (22) and F distribution | Properties, proofs, exercises (23).

Proposition The probability density function of F distribution | Properties, proofs, exercises (24) can be written asF distribution | Properties, proofs, exercises (25)where:

  1. F distribution | Properties, proofs, exercises (26) is the probability density function of a Gamma random variable with parameters F distribution | Properties, proofs, exercises (27) and F distribution | Properties, proofs, exercises (28):F distribution | Properties, proofs, exercises (29)

  2. F distribution | Properties, proofs, exercises (30) is the probability density function of a Gamma random variable with parameters F distribution | Properties, proofs, exercises (31) and F distribution | Properties, proofs, exercises (32):F distribution | Properties, proofs, exercises (33)

Proof

We need to prove thatF distribution | Properties, proofs, exercises (34)whereF distribution | Properties, proofs, exercises (35)andF distribution | Properties, proofs, exercises (36)Let us start from the integrand function: F distribution | Properties, proofs, exercises (37)where F distribution | Properties, proofs, exercises (38)and F distribution | Properties, proofs, exercises (39) is the probability density function of a random variable having a Gamma distribution with parameters F distribution | Properties, proofs, exercises (40) and F distribution | Properties, proofs, exercises (41). Therefore,F distribution | Properties, proofs, exercises (42)

Relation to the Chi-square distribution

In the introduction, we have stated (without a proof) that a random variable F distribution | Properties, proofs, exercises (43) has an F distribution with F distribution | Properties, proofs, exercises (44) and F distribution | Properties, proofs, exercises (45) degrees of freedom if it can be written as a ratioF distribution | Properties, proofs, exercises (46)where:

  1. F distribution | Properties, proofs, exercises (47) is a Chi-square random variable with F distribution | Properties, proofs, exercises (48) degrees of freedom;

  2. F distribution | Properties, proofs, exercises (49) is a Chi-square random variable, independent of F distribution | Properties, proofs, exercises (50), with F distribution | Properties, proofs, exercises (51) 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): F distribution | Properties, proofs, exercises (52) can be thought of as a Gamma random variable with parameters F distribution | Properties, proofs, exercises (53) and F distribution | Properties, proofs, exercises (54), where the parameter F distribution | Properties, proofs, exercises (55) is equal to the reciprocal of another Gamma random variable F distribution | Properties, proofs, exercises (56), independent of the first one, with parameters F distribution | Properties, proofs, exercises (57) and F distribution | Properties, proofs, exercises (58). The equivalence can be proved as follows.

Since a Gamma random variable with parameters F distribution | Properties, proofs, exercises (59) and F distribution | Properties, proofs, exercises (60) is just the product between the ratio F distribution | Properties, proofs, exercises (61) and a Chi-square random variable with F distribution | Properties, proofs, exercises (62) degrees of freedom (see the lecture entitled Gamma distribution), we can write F distribution | Properties, proofs, exercises (63)where F distribution | Properties, proofs, exercises (64) is a Chi-square random variable with F distribution | Properties, proofs, exercises (65) degrees of freedom. Now, we know that F distribution | Properties, proofs, exercises (66) is equal to the reciprocal of another Gamma random variable F distribution | Properties, proofs, exercises (67), independent of F distribution | Properties, proofs, exercises (68), with parameters F distribution | Properties, proofs, exercises (69) and F distribution | Properties, proofs, exercises (70). Therefore,F distribution | Properties, proofs, exercises (71)But a Gamma random variable with parameters F distribution | Properties, proofs, exercises (72) and F distribution | Properties, proofs, exercises (73) is just the product between the ratio F distribution | Properties, proofs, exercises (74) and a Chi-square random variable with F distribution | Properties, proofs, exercises (75) degrees of freedom. Therefore, we can write F distribution | Properties, proofs, exercises (76)

Expected value

The expected value of an F random variable F distribution | Properties, proofs, exercises (77) is well-defined only for F distribution | Properties, proofs, exercises (78) and it is equal toF distribution | Properties, proofs, exercises (79)

Proof

It can be derived thanks to the integral representation of the Beta function:F distribution | Properties, proofs, exercises (80)

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 F distribution | Properties, proofs, exercises (81): when F distribution | Properties, proofs, exercises (82), 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 F distribution | Properties, proofs, exercises (83) is well-defined only for F distribution | Properties, proofs, exercises (84) and it is equal toF distribution | Properties, proofs, exercises (85)

Proof

It can be derived thanks to the usual variance formula (F distribution | Properties, proofs, exercises (86)) and to the integral representation of the Beta function:F distribution | Properties, proofs, exercises (87)

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 F distribution | Properties, proofs, exercises (88): when F distribution | Properties, proofs, exercises (89), the above improper integrals do not converge (both arguments of the Beta function must be strictly positive).

Higher moments

The F distribution | Properties, proofs, exercises (90)-th moment of an F random variable F distribution | Properties, proofs, exercises (91) is well-defined only for F distribution | Properties, proofs, exercises (92) and it is equal toF distribution | Properties, proofs, exercises (93)

Proof

It is obtained by using the definition of moment:F distribution | Properties, proofs, exercises (94)

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 F distribution | Properties, proofs, exercises (95): when F distribution | Properties, proofs, exercises (96), the above improper integrals do not converge (both arguments of the Beta function must be strictly positive).

Moment generating function

An F random variable F distribution | Properties, proofs, exercises (97) does not possess a moment generating function.

Proof

When a random variable F distribution | Properties, proofs, exercises (98) possesses a moment generating function, then the F distribution | Properties, proofs, exercises (99)-th moment of F distribution | Properties, proofs, exercises (100) exists and is finite for any F distribution | Properties, proofs, exercises (101). But we have proved above that the F distribution | Properties, proofs, exercises (102)-th moment of F distribution | Properties, proofs, exercises (103) exists only for F distribution | Properties, proofs, exercises (104). Therefore, F distribution | Properties, proofs, exercises (105) 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 isF distribution | Properties, proofs, exercises (106)where the integralF distribution | Properties, proofs, exercises (107)is known as incomplete Beta function and is usually computed numerically with the help of a computer algorithm.

Proof

This is proved as follows:F distribution | Properties, proofs, exercises (108)

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 F distribution | Properties, proofs, exercises (109) and F distribution | Properties, proofs, exercises (110);

  • the orange line is the pdf of an F random variable with parameters F distribution | Properties, proofs, exercises (111) and F distribution | Properties, proofs, exercises (112).

By increasing the first parameter from F distribution | Properties, proofs, exercises (113) to F distribution | Properties, proofs, exercises (114), 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.

F distribution | Properties, proofs, exercises (115)

Plot 2 - Increasing the second parameter

In the following plot:

  • the blue line is the density of an F distribution with parameters F distribution | Properties, proofs, exercises (116) and F distribution | Properties, proofs, exercises (117);

  • the orange line is the density of an F distribution with parameters F distribution | Properties, proofs, exercises (118) and F distribution | Properties, proofs, exercises (119).

By increasing the second parameter from F distribution | Properties, proofs, exercises (120) to F distribution | Properties, proofs, exercises (121), the mean of the distribution (vertical line) decreases (from F distribution | Properties, proofs, exercises (122) to F distribution | Properties, proofs, exercises (123)) and some density is shifted from the tails (mostly from the right tail) to the center of the distribution.

F distribution | Properties, proofs, exercises (124)

Plot 3 - Increasing both parameters

In the next plot:

  • the blue line is the density of an F random variable with parameters F distribution | Properties, proofs, exercises (125) and F distribution | Properties, proofs, exercises (126);

  • the orange line is the density of an F random variable with parameters F distribution | Properties, proofs, exercises (127) and F distribution | Properties, proofs, exercises (128).

By increasing the two parameters, the mean of the distribution decreases (from F distribution | Properties, proofs, exercises (129) to F distribution | Properties, proofs, exercises (130)) 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.

F distribution | Properties, proofs, exercises (131)

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let F distribution | Properties, proofs, exercises (132) be a Gamma random variable with parameters F distribution | Properties, proofs, exercises (133) and F distribution | Properties, proofs, exercises (134).

Let F distribution | Properties, proofs, exercises (135) be another Gamma random variable, independent of F distribution | Properties, proofs, exercises (136), with parameters F distribution | Properties, proofs, exercises (137) and F distribution | Properties, proofs, exercises (138).

Find the expected value of the ratioF distribution | Properties, proofs, exercises (139)

Solution

We can writeF distribution | Properties, proofs, exercises (140)where F distribution | Properties, proofs, exercises (141) and F distribution | Properties, proofs, exercises (142) are two independent Gamma random variables, the parameters of F distribution | Properties, proofs, exercises (143) are F distribution | Properties, proofs, exercises (144) and F distribution | Properties, proofs, exercises (145) and the parameters of F distribution | Properties, proofs, exercises (146) are F distribution | Properties, proofs, exercises (147) and F distribution | Properties, proofs, exercises (148) (see the lecture entitled Gamma distribution). By using this fact, the ratio can be written asF distribution | Properties, proofs, exercises (149)where F distribution | Properties, proofs, exercises (150) has an F distribution with parameters F distribution | Properties, proofs, exercises (151) and F distribution | Properties, proofs, exercises (152). Therefore,F distribution | Properties, proofs, exercises (153)

Exercise 2

Find the third moment of an F random variable with parameters F distribution | Properties, proofs, exercises (154) and F distribution | Properties, proofs, exercises (155).

Solution

We need to use the formula for the F distribution | Properties, proofs, exercises (156)-th moment of an F random variable:F distribution | Properties, proofs, exercises (157)

Plugging in the parameter values, we obtainF distribution | Properties, proofs, exercises (158)where 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.

F distribution | Properties, proofs, exercises (2024)
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