Correlation coefficient: show 1-r^2 is the ratio of 0th and 1st order models

[applestrudle]

Homework Statement

You have a linear model y = a+bx. Using the mean square error function for a zeroth order model (b=0 and a = <y>) and a first order model b=Covariance(x,y)/Variance(x) and a = <y> - b<x> show that E1/E0 = 1-r^2

Relevant Equations

MSE function E = <(y - a -bx)^2>
Correlation coefficient r = Covariance(x,y)/Standardev(x)Standarddev(y)
Standarddev = Square root of variance

The zeroth order model gives E0 = Var(y)

I've tried two methods:
Calculating 1-r^2 and trying to get E1/E0.
Calculating E1/E0 and trying to get 1-r^2.

 

[mjc123]

I've tried two methods:

And what do you get?

 

[WWGD]

@applestrudle I'm a bit confused by your notation overall. You defined E as a function of x ( equiv y) then used E0, E1. Is E0:=E(0), E1:=E(1)?

 

[Office_Shredder]

You haven't written down what $r^2$ is, which feels like an important piece.