Property of correlation coefficient

[Max.Planck]

Homework Statement

Prove the following:
Let \rho be the correlation coefficient.
Prove:
\rho(X, Y) = 1 \iff P(Y=aX+b) = 1

Homework Equations

\rho(X, Y) = cov(X,Y)/\sigma_x\sigma_y

The Attempt at a Solution

I have no idea how to prove this, I know that rho must lie between 1 and -1 (inclusive) and that values close to 1 indicate that high values of X must go with high values of Y. But I don't know how to formally prove the above problem.

 

[Max.Planck]

bump. I can't believe nobody hasn't proven or seen the proof of this.

 

[lanedance]

As this is an if and only if, you must prove both ways, I suggest starting with
P(Y=aX+b)=1 \ \ \implies \ \ \rho(X,Y) = 1

Now as the probability is one, you can start with
Y=aX+b

Now assume you know \mu_X, \sigma_x and calculate \rho(X,Y)

The other direction may be slightly trickier, but you should get some good insights from the first exercise