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STA260 Lecture 01

STA260 Lecture 01 Raw

• Overview

  • Source Material: STA260H5S_Winter2026_Chapter6.pdf
  • Topics: Functions of a Random Sample, Point Estimation, Sampling Distributions.
  • Key Theorems derived:
    • Distribution of linear combinations of Normal variables.
    • Sampling distribution of $\overline{X}$ (Normal population).
    • Unbiasedness of Sample Variance.

• Core Concepts

  • Random Sample: Independent and identically distributed (iid) variables.
  • Linear Combination of Independent Normal Variables:
    • If $X_i$ are independent Normal, their linear combination is Normal.
    • Used to prove the distribution of $\overline{X}$.
  • Sampling Distribution of the Sample Mean:
    • $\overline{X}\sim N(\mu ,{\sigma }^2/n)$ given a Normal population.
  • Expectation and Variance of Sample Mean:
    • Proofs that $E[\overline{X}]=\mu$ and $Var(\overline{X})={\sigma }^2/n$ regardless of distribution shape.
  • Unbiasedness of Sample Variance:
    • Proof that $E[S^2]={\sigma }^2$.

• Related Distributions (Pre-Lecture)

  • Chi-squared Distribution: Sum of squared standard normals; models sample variance.
  • T Distribution: Sampling distribution of mean when $\sigma$ is unknown.
  • F Distribution: Ratio of two independent Chi-squared variables; compares variances.

• Footnotes