F-distribution

Definition:

• Related to Chi-squared distribution
• Usually used with ratios of variances
• Let $U_1\sim {\chi }^2({\nu }_1)$ and $U_2\sim {\chi }^2({\nu }_2)$ , Chi-squared distribution
• Then $\frac{U_1/{\nu }_1}{U_2/{\nu }_2}\sim F({\nu }_1,{\nu }_2)$, has 2 dof
  • $F(a,b)\ne F(b,a)$ for $a\ne b$
• Right-skewed
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Mean of F-distribution:

• $\mu =\frac{{\nu }_2}{{\nu }_2-2}$ for ${\nu }_2>2$

Median of $F$-distribution:

• Tends to 1 when ${\nu }_1={\nu }_2$