Sampling Distributions

Learning Outcomes

• Covariance Properties

• Sampling Distributions

• Central Limit Theorem

Covariance

Covariance

Observing Data

Observing Data

Descriptive Statistics

Descriptive Statistics

Sample Mean

Sample Variance

Properties

Expected Value

Variance

Sampling Distributions

Sampling Mean

Sampling Mean

Central Limit Theorem

Central Limit Theorem

Let \(X_1, X_2, \ldots, X_n\) be identical and independent distributed random variables with \(E(X_i)=\mu\) and \(Var(X_i) = \sigma²\). We define \[ Y_n = \sqrt n \left(\frac{\bar X-\mu}{\sigma}\right) \mathrm{ where }\ \bar X = \frac{1}{n}\sum^n_{i=1}X_i. \]

Then, the distribution of the function \(Y_n\) converges to a standard normal distribution function as \(n\rightarrow \infty\).