I can't find the product moment coefficient of correlation?

Question

These are the equations of least square regression lines:

$ Y = 20.8 - 0.219 X $ ($Y$ on $ X$)

$X = 16.2 - 0.785 Y$ ($X$ on $Y$)

Find the coefficient of correlation $r$.

Answer 1

Assuming you are talking about regression of one random variable on another (as opposed to finite sample approximations), the regression coefficient of $Y$ on $X$ is $E(XY)/E(X^2)$; that of $X$ on $Y$ is (by symmetry) $E(XY)/E(Y^2)$. So their product is $E(XY)^2/(E(X^2)E(Y^2))$.

A similar formula holds in the finite sample case, where you have expressions like $\sum x_i y_i$, and so on.