# Correlation of two normal distributions

I have a population $x$ with $\sigma/\mu=a$, and a second population $y$ with $\sigma/\mu=b$. The linear regression is $y=cx$. How would I find the correlation between $y$ and $x$, given that the populations are homoscedastic and normal?

If they were independent, I know the correlation would be zero, but I tried putting some sample data in Excel and got a numerical answer. What would be the general way to solve this?

The equation $y = cx$ should be $$\frac y {\sigma_Y} = \rho \frac x {\sigma_X}$$ which becomes $$\frac y {\mu a} = \rho \frac x {\mu b}.$$ So $$c = \rho\frac a b.$$ Thus $\rho = \dfrac{cb} a.$
If you run a linear regression $y=cx+e$ (only one independent variable) and you obtain the $R^2$ coefficient, its square root $\sqrt{R^2}$ is equal to the absolute value of the sample correlation between $x$ and $y$.
• Well, of course, it can be negative. The square root of the $R^2$ is the absolute value of the correlation. – Anna SdTC Mar 27 '17 at 20:59