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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?

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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 $\rho = \dfrac{cb} a.$

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    $\begingroup$ Thanks, I should have realized that the correlation could be directly calculated from the regression in this case. $\endgroup$ – yunyu Feb 9 '17 at 5:37
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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 $y$.

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  • $\begingroup$ --- or its negative! $\endgroup$ – kjetil b halvorsen Mar 27 '17 at 18:21
  • $\begingroup$ Well, of course, it can be negative. The square root of the $R^2$ is the absolute value of the correlation. $\endgroup$ – Anna SdTC Mar 27 '17 at 20:59
  • $\begingroup$ Yes, can you edit your answer such? $\endgroup$ – kjetil b halvorsen Mar 27 '17 at 21:11

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