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)

FAQs

What is the equation for the F-distribution? ›

5 The F Distribution

Then, the ratio F = ( V / υ 1 ) / ( W / υ 2 ) is an F distribution with ν1 d.f. in the numerator and ν2 d.f. in the denominator. It is usually abbreviated as F ν 1 , ν 2 .

How is the F-distribution derived? ›

The F distribution is derived from the Student's t-distribution. The values of the F distribution are squares of the corresponding values of the t-distribution. One-Way ANOVA expands the t-test for comparing more than two groups.

What does an F-distribution look like? ›

The graph of the F distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom.

What are the 7 properties of normal distribution? ›

Normal distributions are symmetric, unimodal, and asymptotic, and the mean, median, and mode are all equal. A normal distribution is perfectly symmetrical around its center. That is, the right side of the center is a mirror image of the left side. There is also only one mode, or peak, in a normal distribution.

What are the three properties of distribution function? ›

There are three basic properties of a distribution: location, spread, and shape. The location refers to the typical value of the distribution, such as the mean. The spread of the distribution is the amount by which smaller values differ from larger ones.

How to calculate probability of F distribution? ›

Select a random sample of size n1 from a normal population, having a standard deviation equal to σ1. Select an independent random sample of size n2 from a normal population, having a standard deviation equal to σ2. The f statistic is the ratio of s1212 and s2222. Thus, f = [ s1212 ] / [ s2222]

What is the formula for the F distribution test? ›

The f test statistic formula is given below: F statistic for large samples: F = σ21σ22 σ 1 2 σ 2 2 , where σ21 σ 1 2 is the variance of the first population and σ22 σ 2 2 is the variance of the second population.

What is skewness and kurtosis of F distribution? ›

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution.

What is an example of the F distribution in real life? ›

The F-distribution has numerous real-world applications. For example, it is used in finance to test whether the variances of stock returns are equal across two or more portfolios. It is also used in engineering to test the effectiveness of different manufacturing processes by comparing the variances of the outcomes.

What is the difference between chi-square and F distribution? ›

A chi-square distribution is defined by one parameter (i.e., n-1 degrees of freedom), while an F-distribution is defined by parameters, i.e., degrees of freedom of the numerator (m) and degrees of freedom of the denominator (n).

What is the F distribution also known as? ›

In probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W.

How do you know when to use an F distribution? ›

The 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.

What is the decision rule for the F distribution? ›

Decision Rule: Determine the critical value of F from the F-distribution tables for a chosen significance level (α, often 0.05) and the degrees of freedom from both samples. If the calculated F-statistic is greater than the critical value from the F-distribution table, reject H0.

What does the F distribution always range from? ›

The F distribution is always strongly skewed left. The values of the F statistic range from to −∞ to ∞. The F distribution can only be used for one-sided hypothesis tests. The F test statistic must always be a positive number.

What are the properties of the F-test? ›

Due to such relationships, the F-test has many properties, like chi square. The F-values are all non negative. The F-distribution in the F-test is always non-symmetrically distributed. The mean in F-distribution in the F-test is approximately one.

What are the three characteristics of the sampling distribution of F? ›

What are three characteristics of the sampling distribution of F ? 1) It is positively skewed. 2) It is always a positive value. 3) Its median value is approximately 1.

What are the properties of F transform? ›

Fourier Transform Properties
  • Duality – If h(t) has a Fourier transform H(f), then the Fourier transform of H(t) is H(-f).
  • Linear transform – Fourier transform is a linear transform. ...
  • Modulation property – According to the modulation property, a function is modulated by the other function, if it is multiplied in time.

Which of the following is a characteristic of the f distribution? ›

Answer and Explanation:

The distribution is (c) positively skewed. As a ratio of two positive quantities, the distribution can take only positive values, with no upper limit, meaning it is positively skewed.

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