Property of correlation coefficient

1. Mar 18, 2012

Max.Planck

1. The problem statement, all variables and given/known data
Prove the following:
Let $$\rho$$ be the correlation coefficient.
Prove:
$$\rho(X, Y) = 1 \iff P(Y=aX+b) = 1$$

2. Relevant equations
$$\rho(X, Y) = cov(X,Y)/\sigma_x\sigma_y$$

3. 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.

2. Mar 19, 2012

Max.Planck

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

3. Mar 19, 2012

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