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  • STA260 Lecture 06

STA260 Lecture 06 Raw

• Review
  • Standardizing
  • $Z=\frac{X-\mu }{\sigma }\sim N(0,1)$
  • $Z=\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}\sim N(0,1)$
  • $\sum _{i=1}^n{Z}_i^2$
    • Then each $Z_i\stackrel{iid}{\sim }N(0,1)$
    • $Z_i^2\sim {\chi }_{(n)}^2$
    • Chi-squared from Sum of Squared Standard Normals
    • Sum of Squares is Chi-squared
• T Distribution
• T Distribution Example. Show it's a T dist.
  • $\frac{\overline{X}-\mu }{\frac{S}{\sqrt{n}}}\sim t_{(n-1)}$
  • Since $T=\frac{Z}{\sqrt{\frac{{\chi }_{(\nu )}^2}{\nu }}}$
  • Distribution of S^2
  • $\frac{\sqrt{n}(\overline{X}-\mu )}{S}$
  • Divide top and bottom by $\sigma$
  • $\frac{\frac{\sqrt{n}(\overline{X}-\mu )}{\sigma }}{\frac{S}{\sigma }}$
  • $\frac{\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}}{\frac{S}{\sigma }}$
  • Square and root the bottom
  • $\frac{\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}}{\sqrt{\frac{S^2}{{\sigma }^2}}}$
  • Multiply and divide by $(n-1)$
  • $\frac{\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}}{\sqrt{\frac{\frac{(n-1)S^2}{{\sigma }^2}}{(n-1)}}}$
  • Top is $Z\sim N(0,1)$
  • Bottom is ${\chi }_{(n-1)}^2$
  • Divided by $n-1$
  • $\frac{Z}{\sqrt{\frac{U}{n-1}}}$
  • So it's $t_{(n-1)}$
• F Distribution
  • We use $Z=\frac{X-\mu }{\sigma }$ or $\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}\sim N(0,1)$ to calculate probabilities.
  • $\frac{(n-1)S^2}{{\sigma }^2}\sim {\chi }_{(n-1)}^2$, the Marginal Distribution of Sample Variance, when we want to find probability relating to sample variance.
  • $\frac{\overline{X}-\mu }{\frac{S}{\sqrt{n}}}\sim t_{(n-1)}$ is used to calculate probabilities when $\sigma$ is not known.