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  • STA260 Lecture 04

F Distribution

• Ratio of two independent Random Variables from Chi-squared Distribution adjusted by their degrees of freedom.
• Use: Questions involving a ratio between two Sample Variances.
• $F=\frac{\frac{U_1}{n_1}}{\frac{U_2}{n_2}}\sim F_{n_1,n_2}$
• $U$ and $V$ are independent random variables from ${\chi }_{(m)}^2$ and ${\chi }_{(n)}^2$ respectively.
• We then have $F=\frac{\frac{U}{m}}{\frac{V}{n}}\sim F_{m,n}$
• Left-tail:
  • $F_{1-\alpha ,m,n}=\frac{1}{F_{\alpha ,n,m}}$
• Proof:
  • $X_1,\dots ,X_m\stackrel{iid}{\sim }N({\mu }_1,{\sigma }_1^2)$
  • $Y_1,\dots ,Y_n\stackrel{iid}{\sim }N({\mu }_2,{\sigma }_2^2)$
  • They're independent since $U\sim {\chi }_{(m-1)}^2$ and $V\sim {\chi }_{(n-1)}^2$
  • $\frac{\frac{S_1^2}{{\sigma }_1^2}}{\frac{S_2^2}{{\sigma }_2^2}}=\frac{\frac{\frac{(m-1)S_1^2}{{\sigma }_2^2}}{(m-1)}}{\frac{\frac{(n-1)S_2^2}{{\sigma }_2^2}}{(n-1)}}=\frac{\frac{{\chi }_{(m-1)}^2}{m-1}}{\frac{{\chi }_{(n-1)}^2}{n-1}}=\frac{\frac{U}{m-1}}{\frac{V}{n-1}}\sim F_{m-1,n-1}$
• Example (slide 29):
  • Independent samples of $m=6$ and $n=10$
    • We have $X_1,\dots ,X_6\sim N({\mu }_X,{\sigma }_1^2)$
    • We have $Y_1,\dots ,Y_{10}\sim N({\mu }_Y,{\sigma }_2^2)$
  • Both from Normal Distribution
  • ${\sigma }_1^2={\sigma }_2^2$
  • Find $P(\frac{S_1^2}{S_2^2}\le b)=0.95$
  • ${\displaystyle \frac{\frac{S_1^2}{{\sigma }_1^2}}{\frac{S_2^2}{{\sigma }_2^2}}}=\frac{S_1^2}{S_2^2}\sim F_{5,9}$
    • We can stop after saying $\frac{\frac{S_1^2}{{\sigma }_1^2}}{\frac{S_2^2}{{\sigma }_2^2}}$ because of the result we have.
  • Otherwise we can:
    • $\frac{\frac{S_1^2}{{\sigma }_1^2}}{\frac{S_2^2}{{\sigma }_2^2}}$
    • $\frac{\frac{\frac{(m-1)S_1^2}{{\sigma }_1^2}}{m-1}}{\frac{\frac{(n-1)S_2^2}{{\sigma }_2^2}}{n-1}}$
    • $\frac{\frac{{\chi }_{(m-1)}^2}{m-1}}{\frac{{\chi }_{(n-1)}^2}{n-1}}\sim F_{(m-1,n-1)}$
  • We invert it because our table has right-tail values only.
  • $P(F_{(5,9)}>b)=0.05$
    • $0.05=\alpha$
    • $F_{(0.05;5,9)}=3.48$
  • $3.48$ on the f table.
  • F critical value lookup (F_5,9 example)