STA260 Lecture 01

Pre-Lecture

STA260 Pre-Lecture 01 Summary
STA260H5S_Winter2026_Chapter6.pdf

Lecture

STA260H5S_Winter2026_Chapter6.pdf

6.3
- Random Sample can also mean Independent and identically distributed.
- Functions of a Random Sample
- Linear Combinations of Random Variables#STA260
- Sample Mean and Sample Variance
6.2
- Point Estimators
- Confidence Intervals
- Hypothesis Tests
- Distribution of $\bar{X}$
  - A population's distribution is a Normal Distribution
  - Let $X_{1}, \dots, X_{n}$ be independent random variables from a Normal Distribution with $E [X_{i}] = μ_{i}$ and $Var (X_{i}) = σ_{i}^{2}$
  - Then $Y = \sum_{i = 1}^{n} a_{i} X_{i} \sim N (\sum_{i = 1}^{n} a_{i} μ_{i}, \sum_{i = 1}^{n} a_{i}^{2} σ_{i}^{2})$
  - If they're iid from a Random Sample, from $N (μ, σ^{2})$ then:
    - $\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i} \sim N (μ, \frac{σ^{2}}{n})$
  - Proof:
    - Let $X_{1}, \dots, X_{n}$ be independent random variables from a Normal Distribution.
    - $E [X_{i}] = μ_{i}$
    - $Var (X_{i}) = σ_{i}^{2}$
    - $Y = \sum_{i = 1}^{n} a_{i} X_{i}$
    - $Y \sim N (\sum_{i = 1}^{n} a_{i} μ_{i}, \sum_{i = 1}^{n} a_{i}^{2} σ_{i}^{2})$
    - Using the Moment Generating Function of Y:
      - $M_{Y} (t) = E [e^{t Y}] = E [e^{t (\sum_{i = 1}^{n} a_{i} X_{i})}]$
      - $= E [e^{t a_{1} X_{1}} e^{t a_{2} X_{2}} \dots e^{t a_{n} X_{n}}]$
      - Because the $X_{i}$ are independent, the expectation factors:
      - $= E [e^{t a_{1} X_{1}}] E [e^{t a_{2} X_{2}}] \dots E [e^{t a_{n} X_{n}}]$
      - $= M_{X_{1}} (t a_{1}) M_{X_{2}} (t a_{2}) \dots M_{X_{n}} (t a_{n})$
      - For a normal $X_{i}$ ,
      - $M_{X_{i}} (t) = e^{μ_{i} t + σ_{i}^{2} \frac{t^{2}}{2}} t \in R$
      - This is the MGF of a Normal Distribution
      - $= M_{Y} (t) = e^{a_{1} μ_{1} t + σ_{1}^{2} a_{1} \frac{t^{2}}{2}} \dots e^{a_{n} μ_{n} t + σ_{n}^{2} a_{n}^{2} \frac{t^{2}}{2}}$
      - $= e^{(\sum_{i = 1}^{n} a_{i} μ_{i}) t + (\sum_{i = 1}^{n} a_{i}^{2} σ_{i}^{2}) \frac{t^{2}}{2}}$
      - This is the MGF of $N (\sum_{i = 1}^{n} a_{i} μ_{i}, \sum_{i = 1}^{n} a_{i}^{2} σ_{i}^{2})$ so then $Y$ has that Normal Distribution.

Post-Lecture

Sample Mean proof:
- Let $X_{1}, \dots, X_{n}$ be a Random Sample from a distribution with $E [X_{i}] = μ$ and $Var (X_{i}) = σ^{2}$
- Define the sample mean as $\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}$
- Then $E [\bar{X}] = μ$ and $Var (\bar{X}) = \frac{σ^{2}}{n}$
- Proof:
  - Want to show that $E [\bar{X}] = μ$
  - $\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}$
  - $E [\bar{X}] = E [\frac{1}{n} \sum_{i = 1}^{n} X_{i}]$
  - Since expectation is linear
  - $E [\bar{X}] = \frac{1}{n} E [\sum_{i = 1}^{n} X_{i}]$
  - $E [\bar{X}] = \frac{1}{n} E [X_{1} + X_{2} + \dots + X_{n}]$
  - $E [\bar{X}] = \frac{1}{n} [E [X_{1}] + E [X_{2}] + \dots + E [X_{n}]]$
  - $E [\bar{X}] = \frac{1}{n} [\underset{n times}{\underset{⏟}{μ + μ + \dots + μ}}]$
  - $E [\bar{X}] = \frac{1}{n} n μ = μ$
- Proof:
  - Want to show that $Var (\bar{X}) = \frac{σ^{2}}{n}$
  - $\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}$
  - $Var (\bar{X}) = Var (\frac{1}{n} \sum_{i = 1}^{n} X_{i})$
  - Linearity of variance
  - $Var (\bar{X}) = \frac{1}{n^{2}} Var (\sum_{i = 1}^{n} X_{i})$
  - $Var (\bar{X}) = \frac{1}{n^{2}} Var (X_{1} + X_{2} + \dots + X_{n})$
  - Lemma: $Var (X_{1} + X_{2} + \dots + X_{n}) = n σ^{2}$
    - $Var (X_{1}) + Var (X_{2}) + \dots + Var (X_{n})$
    - $σ^{2} + σ^{2} + \dots + σ^{2}$ n times
    - $n σ^{2}$
  - $Var (\bar{X}) = \frac{1}{n^{2}} n σ^{2} = \frac{σ^{2}}{n}$
Sample Variance proof:
- Let $X_{1}, \dots, X_{n}$ be a Random Sample from a distribution with $E [X_{i}] = μ$ and $Var (X_{i}) = σ^{2}$
- Then $E [S^{2}] = σ^{2}$
- $S^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - \bar{X})^{2}$
- $S^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i}^{2} - 2 X_{i} \bar{X} - {\bar{X}}^{2})$
- $E [S^{2}] = E [\frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i}^{2} - 2 X_{i} \bar{X} - {\bar{X}}^{2})]$
- $E [S^{2}] = \frac{1}{n - 1} E [\sum_{i = 1}^{n} (X_{i}^{2} - 2 X_{i} \bar{X} - {\bar{X}}^{2})]$
- Lemma: $E [\sum_{i = 1}^{n} (X_{i}^{2} - 2 X_{i} \bar{X} - {\bar{X}}^{2})] = (n - 1) σ^{2}$
  - $E [\sum_{i = 1}^{n} X_{i}^{2} - 2 \sum_{i = 1}^{n} X_{i} \bar{X} - \sum_{i = 1}^{n} {\bar{X}}^{2}]$
  - $E [\sum_{i = 1}^{n} X_{i}^{2} - 2 \bar{X} \sum_{i = 1}^{n} X_{i} - \sum_{i = 1}^{n} {\bar{X}}^{2}]$
  - $\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}$
  - $E [\sum_{i = 1}^{n} X_{i}^{2} - \frac{2}{n} \sum_{i = 1}^{n} X_{i} \sum_{i = 1}^{n} X_{i} - \sum_{i = 1}^{n} {\bar{X}}^{2}]$
  - $E [\sum_{i = 1}^{n} X_{i}^{2} - \frac{2}{n} \sum_{i = 1}^{n} X_{i}^{2} - \sum_{i = 1}^{n} {\bar{X}}^{2}]$
  - $E [\frac{n}{n} \sum_{i = 1}^{n} X_{i}^{2} - \frac{2}{n} \sum_{i = 1}^{n} X_{i}^{2} - \sum_{i = 1}^{n} {\bar{X}}^{2}]$
  - $E [\frac{n - 2}{n} \sum_{i = 1}^{n} X_{i}^{2} - \sum_{i = 1}^{n} {\bar{X}}^{2}]$
  - $\frac{n - 2}{n} E [\sum_{i = 1}^{n} X_{i}^{2}] - E [\sum_{i = 1}^{n} {\bar{X}}^{2}]$
  - $\frac{n - 2}{n} (E [X_{1}^{2}] + \dots + E [X_{n}^{2}]) - n E [{\bar{X}}^{2}]$
  - $Var (X_{i}) = E [X_{i}^{2}] - E [X_{i}]^{2}$
  - $Var (X_{i}) + E [X_{i}]^{2} = E [X_{i}^{2}]$
  - $Var (\bar{X}) = E [{\bar{X}}^{2}] - E [\bar{X}]^{2}$
  - $Var (\bar{X}) + E [\bar{X}]^{2} = E [{\bar{X}}^{2}]$
  - $\frac{n - 2}{n} (Var (X_{1}) + E [X_{1}]^{2} + \dots + Var (X_{n}) + E [X_{n}]^{2}) - n [Var (\bar{X}) + E [\bar{X}]^{2}]$
  - $\frac{n - 2}{n} (σ^{2} + μ^{2} + \dots + σ^{2} + μ^{2}) - n [\frac{σ^{2}}{n} + μ^{2}]$
  - $(n - 2) (σ^{2} + μ^{2}) - n [\frac{σ^{2}}{n} + μ^{2}]$
  - $(n - 2) (σ^{2} + μ^{2}) - σ^{2} + n μ^{2}$
  - $n σ^{2} + n μ^{2} - 2 σ^{2} - 2 μ^{2} - σ^{2} + n μ^{2}$
  - $n σ^{2} + 2 n μ^{2} - 3 σ^{2} - 2 μ^{2}$
  - #tk