- #1

stunner5000pt

- 1,461

- 2

## Homework Statement

Why is the correlation coefficient between -1 and +1?

## Homework Equations

we know correlation coefficient

[tex] \rho = \frac{E[xy]-E[x]E[y]}{\sqrt{\sigma_{x} \sigma_{y}}} [/tex]

OR

[tex] r = \frac{\sum ^n _{i=1}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum ^n _{i=1}(X_i - \bar{X})^2} \sqrt{\sum ^n _{i=1}(Y_i - \bar{Y})^2}} [/tex]

## The Attempt at a Solution

Is there a way to prove this analytically? Perhaps we can use the second formula and prove by induction the bottom is greater than the top? Or perhaps equal??

I tried using the expected value formula for the first version with rho - i couldn't really use that properly. Canyou please suggest an approach? Can this even be done analytically? Or would it just have to be explained?

Thanks for your help!