1. Statistics 350 Lecture 14

2. Today Last Day: Matrix results and Chapter 5 Today: More matrix results and Chapter 5 Please read Chapter 5

3. Multivariate Normal Distribution Suppose have a random vector Y, with elements Y i E(Y)=  2 (Y)=  =

4. Multivariate Normal Distribution Suppose have a random vector Y, with elements Y i Y is said to have a multivariate normal distribtuion if it has density The density for each Y i is

5. Multivariate Normal Distribution and Simple Linear Regression For the simple linear regression model, Y has a multivariate normal distribution:

6. Multivariate Normal Distribution and Simple Linear Regression Distribution of b:

7. Multivariate Normal Distribution and Simple Linear Regression Suppose we could choose the X i ’s where we observe the Y’s What does covariance matrix tell us about the choice of X’s

8. Fitted Values and Residuals Fitted values: Hat matrix: Properties of the hat-matrix

9. Fitted Values and Residuals Residuals: Covariance of residuals Estimated covariance:

10. Fitted Values and Residuals Geometric interpretation of residuals:

11. Fitted Values and Residuals Geometric interpretation of residuals:

12. Analysis of Variance Results SSTO SSE

13. Analysis of Variance Results SSR

14. Analysis of Variance Results Quadratic Forms: