STA260 Lecture 04

Lecture

STA260 Lecture 04 Raw
STA260 Lecture 04 Flashcards

Core Ideas

• Standardize a normal Random Sample to get iid Standard Normal Distribution variables.
• Squaring and summing standardized normals produces a Chi-squared Distribution.
• The Sample Variance (scaled by the true variance) has a chi-squared distribution under normal sampling.

From Normal Sample to Chi-squared

• Let $X_1,\dots ,X_n$ be a Random Sample from $N(\mu ,{\sigma }^2)$.
• Define $Z_i=\frac{X_i-\mu }{\sigma }$.
• Then $Z_i\stackrel{iid}{\sim }N(0,1)$.
• Then $Z_i^2\sim {\chi }_{(1)}^2$.
• Define:
  • $W=\sum _{i=1}^n{Z}_i^2$
  • $=\sum _{i=1}^n{\left(\frac{X_i-\mu }{\sigma }\right)}^2$
  • $\sim {\chi }_{(n)}^2.$
• See:
  • Standardizing a Normal Sample
  • Sum of Squares is Chi-squared
  • Chi-squared Probability Calculations Example

Example: Chi-squared Critical Value

• If $Z_1,\dots ,Z_6\stackrel{iid}{\sim }N(0,1)$, then $\sum _{i=1}^6{Z}_i^2\sim {\chi }_{(6)}^2$.
• Find $b$ such that $P(\sum _{i=1}^6{Z}_i^2\le b)=0.95$.
• This is the 0.95 quantile of ${\chi }_{(6)}^2$ (often written $b={\chi }_{0.95,6}^2$).
• See:
  • Chi-squared Quantiles (Table Lookup)

Marginal Distribution of Sample Variance

• If $X_1,\dots ,X_n$ is a Random Sample from $N(\mu ,{\sigma }^2)$, then:
  • $\frac{(n-1)S^2}{{\sigma }^2}$
  • $=\frac{\sum _{i=1}^n(X_i-\overline{X}{)}^2}{{\sigma }^2}$
  • $\sim {\chi }_{(n-1)}^2$.
• See:
  • Marginal Distribution of Sample Variance

Where This Goes Next

• Leads into:
  • T Distribution
  • F Distribution