F-distribution

F-distribution

The F-distribution, also known as the Fisher-Snedecor distribution, is a probability distribution that arises in statistical inference. It is named after Ronald Fisher and George Snedecor, who independently developed the distribution in the 1920s.

Definition

The F-distribution is a continuous probability distribution that takes on only positive values. It is defined by two positive integer parameters, denoted as "d₁" and "d₂". These parameters represent the degrees of freedom associated with the numerator and denominator of the F-statistic, respectively.

The probability density function (PDF) of the F-distribution is given by the formula:

f(x) = (Γ((d₁ + d₂) / 2) / (Γ(d₁ / 2) * Γ(d₂ / 2))) * (d₁ / d₂)^(d₁ / 2) * x^((d₁ / 2) - 1) * (1 + (d₁ / d₂) * x)^(-(d₁ + d₂) / 2)

where Γ denotes the gamma function.

Properties

The F-distribution has several important properties:

1. Symmetry: The F-distribution is not symmetric. Its shape depends on the values of the degrees of freedom parameters "d₁" and "d₂".

2. Skewness: The F-distribution is positively skewed when "d₁" is less than 2. As "d₁" increases, the distribution becomes more symmetric.

3. Support: The F-distribution is defined for positive values only, as it represents the ratio of two chi-squared random variables.

4. Relationship to other distributions: The F-distribution is related to the chi-squared distribution. Specifically, if "X₁" and "X₂" are independent chi-squared random variables with "d₁" and "d₂" degrees of freedom, respectively, then the ratio "X₁ / X₂" follows an F-distribution with parameters "d₁" and "d₂".

Applications

The F-distribution is widely used in statistical inference, particularly in the analysis of variance (ANOVA) and regression analysis. It plays a crucial role in hypothesis testing and estimation of population variances.

Some common applications of the F-distribution include:

1. ANOVA: The F-test is used to compare the variances of multiple groups in ANOVA. It helps determine if there are significant differences between the means of the groups.

2. Regression analysis: The F-test is used to assess the overall significance of a regression model. It tests whether the regression coefficients are jointly significant.

3. Quality control: The F-distribution is used in quality control to compare the variances of different samples or processes.

References

1. Fisher, R. A. (1925). "Applications of Student's distribution". Metron. 5 (1): 90–104.

2. Snedecor, G. W. (1934). "The distribution of the ratio of the mean square successive difference to the mean square error". Biometrika. 26 (3/4): 404–413.

See also

• Chi-squared distribution
• Analysis of variance
• Regression analysis
• Hypothesis testing
• Estimation theory