Given two regression lines, one for $x$ on $y$ and one for $y$ on $x$, but no actual data points (just the two lines with slopes and intercepts), can I compute Pearson's product moment coefficient $r$?
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Yes.
The slope of one line is $r s_y/s_x$ and the other is $r s_x/s_y$ where $s_y$ is the standard deviation of the $y$'s and $s_x$ is the standard deviation of the $x$'s. From these two slopes, along with the fact that standard deviations are positive, you can easily eliminate the (unknown) standard deviation terms and solve for $r$. Because this sounds like a classroom or textbook question, I will leave you the enjoyment of working out the solution.
Incidentally, the lines must intersect at the point of averages and, once you have computed $r$, you can find $s_y/s_x$. Thus you can obtain four pieces of information about the five lowest bivariate moments of the data (the first and second moments). Clearly that's the best one can do, because the available information is invariant with respect to a uniform rescaling of $(x,y)$ around the point of averages.
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$\begingroup$ A very nice answer (+1). I fear I have given the game away by identifying the duplicate! $\endgroup$ Commented Oct 5, 2015 at 17:35
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$\begingroup$ @Silverfish That's ok. Identifying duplicates is a valuable service; thank you for finding this one. $\endgroup$– whuber ♦Commented Oct 5, 2015 at 17:37
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$\begingroup$ Understanding regression to the mean will answer that question immediately. $\endgroup$– whuber ♦Commented Oct 5, 2015 at 17:53
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$\begingroup$ I have seen that page already but I did not see what I was looking for there $\endgroup$ Commented Oct 5, 2015 at 17:54
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$\begingroup$ Draw a plot showing the two lines and ponder which of them could be exhibiting regression to the mean and which would not. If that's not immediately and perfectly clear, create (or download) a bivariate dataset, compute the two regression lines, and plot those on top of the data. $\endgroup$– whuber ♦Commented Oct 5, 2015 at 18:11