Pearson Product Moment Correlation Coefficient for linear relationship only?

Question

Pearson Product Moment Correlation Coefficient method is used only if variables are linearly correlated.
But if they are linearly correlated, then correlation coefficient $$r=\pm 1$$ only.
Then why we find out r by this method and get something like $r=0.6$?

Answer 1

If two random variables are linearly dependent then you will find $r=\pm 1$. To be linearly dependent means that one is a linear function of the other:

$$Y = a + bX$$

However, you can have a linear correlation without linear dependence, for example

$$Y = a + bX + \epsilon$$

where $\epsilon$ is some other random variable which is independent from $X$. In this case you will find $r^2<1$, and how much less than $1$ depends on the variance of $\epsilon$.