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STA258 Lecture 05

STA258 Lecture 05 Raw
STA258 Lecture 05 Flashcards

• Completed Notes Status

  • Completed insertions: 4 (corrections in variables and subscripts)
  • Ambiguities left unresolved: none

• Lecture Summary

  • Central objective: Establish the consistency of sample estimators and derive the properties of the Chi-squared distribution via standardized normal variables.
  • Key concepts:
    • Consistency: A sequence of estimators is consistent if it converges in probability to the target parameter ($X_n\stackrel{P}{\to }c$) as $n\to \mathrm{\infty }$. This is often proven using Chebyshev's Inequality or by showing bias $\to 0$ and variance $\to 0$.
    • Standardization: Transforming a random variable $Y\sim N(\mu ,{\sigma }^2)$ into $Z=\frac{Y-\mu }{\sigma }\sim N(0,1)$ allows for probability calculations using standard tables.
    • Chi-squared Derivation: The square of a standard normal variable follows a Chi-squared distribution with 1 degree of freedom ($Z^2\sim {\chi }_{(1)}^2$). Consequently, the sum of $n$ squared standard normals follows ${\chi }_{(n)}^2$.
  • Connections:
    • The Weak Law of Large Numbers is a direct application of consistency applied to the sample mean.
    • The consistency of the sample variance ($S^2\stackrel{P}{\to }{\sigma }^2$) relies on the Continuous Mapping Theorem and the Weak Law of Large Numbers.

• TK Resolutions

  • #tk: Show that $S^2\stackrel{P}{\to }{\sigma }^2$
    • Answer: Resolved in the raw note. Two methods were shown: one using the limit of the variance of $S^2$ approaching 0, and another decomposing $S^2$ into components that converge via WLLN and Continuous Mapping Theorem.

• Practice Questions

  • Remember/Understand:
    • What are the mean and variance of a standardized random variable $Z$?
    • If $Z_1,\dots ,Z_n$ are i.i.d. $N(0,1)$, what is the distribution of $\sum Z_i^2$?
    • What condition must the limit of $P(|X_n-X|>ϵ)$ satisfy for convergence in probability?
  • Apply/Analyze:
    • Using the property that $Var(S^2)\to 0$, prove that $S^2$ is a consistent estimator for ${\sigma }^2$.
    • Calculate the sample size $n$ required for the difference of means $\overline{X}-\overline{Y}$ to be within 1 unit of the true difference with 95% probability, given ${\sigma }_1^2=2,{\sigma }_2^2=2.5$.
  • Evaluate/Create:
    • Why is $S^2$ considered consistent despite being a random variable itself? Explain using the concept of degenerate distributions.

• Challenging Concepts

  • Consistency of Sample Variance:
    • Why it's challenging: It involves expanding the summation term and applying limit theorems to multiple components simultaneously.
    • Study strategy: Practice decomposing $S^2$ into $\frac{n}{n-1}(\frac{1}{n}\sum X_i^2-{\overline{X}}^2)$ and applying WLLN to each part separately.
  • Chi-squared as Sum of Normals:
    • Why it's challenging: Remembering the degrees of freedom addition rule and the specific link to $N(0,1)$.
    • Study strategy: Visualize $Z^2$ as a "folded" normal distribution and recall that summing independent ${\chi }^2$ variables adds their degrees of freedom.

• Action Plan

  • Immediate review actions:
    • Review all [COMPLETED] insertions and verify with course materials
    • Address all #tk items (if any)
    • Re-read Lecture Summary and confirm key links
  • Practice and application:
    • Answer Remember/Understand questions
    • Answer Apply/Analyze questions
    • Attempt Evaluate/Create questions
  • Deep dive study:
    • Focus on Challenging Concepts (above)
    • Extend atomic notes only where the new lecture adds content
  • Verification and integration:
    • Cross-reference notes with textbook/slides
    • Identify remaining gaps
    • Link this lecture to prior/future lecture notes (lecture notes only; no MOCs)

• Footnotes