F distribution

The F distribution is a continuous probability distribution that arises frequently in the context of statistical hypothesis testing, particularly in the analysis of variance (ANOVA), the F-test, and the comparison of two variances. It is named after the statistician Ronald Fisher.

Definition

The F distribution is defined as the distribution of the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom. If \( X \sim \chi^2(d_1) \) and \( Y \sim \chi^2(d_2) \) are independent, then the random variable \( F = \frac{X/d_1}{Y/d_2} \) follows an F distribution with \( d_1 \) and \( d_2 \) degrees of freedom.

Probability Density Function

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

\[

f(x; d_1, d_2) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1 + d_2}}}}{x B\left(\frac{d_1}{2}, \frac{d_2}{2}\right)}

\]

where \( B \) is the beta function.

Cumulative Distribution Function

The cumulative distribution function (CDF) of the F distribution is expressed in terms of the incomplete beta function:

\[

F(x; d_1, d_2) = I_{\frac{d_1 x}{d_1 x + d_2}}\left(\frac{d_1}{2}, \frac{d_2}{2}\right)

\]

Properties

Mean

The mean of the F distribution is given by:

\[

\text{Mean} = \frac{d_2}{d_2 - 2}

\]

provided that \( d_2 > 2 \).

Variance

The variance of the F distribution is:

\[

\text{Variance} = \frac{2d_2^2(d_1 + d_2 - 2)}{d_1(d_2 - 2)^2(d_2 - 4)}

\]

for \( d_2 > 4 \).

Mode

The mode of the F distribution is:

\[

\text{Mode} = \frac{d_1 - 2}{d_1} \cdot \frac{d_2}{d_2 + 2}

\]

for \( d_1 > 2 \).

Applications

The F distribution is primarily used in the context of ANOVA and the F-test. It is used to determine whether there are significant differences between the variances of two populations. In ANOVA, it is used to compare the variances across multiple groups to assess whether the means of different groups are significantly different.

Related Distributions

• If \( X \sim F(d_1, d_2) \), then \( \frac{1}{X} \sim F(d_2, d_1) \).
• If \( X \sim F(d_1, d_2) \), then \( \frac{d_1 X}{d_2} \sim \text{Beta}\left(\frac{d_1}{2}, \frac{d_2}{2}\right) \).

See Also

• Chi-squared distribution
• T-distribution
• ANOVA
• F-test