Why is the square root of the product of two regression coefficients, one from X predicting Y and vice versa equal to the correlation?

Question

If there's a regression model where $Y = a + bX$ where $a = 1.6$ and $b=0.4$, it has a correlation coefficient of $r = 0.60302$.

If $X$ and $Y$ are then switched around and the equation becomes $X = c + dY$ where $c=0.4545$ and $d=0.9091$, it also has an $r$ value of $0.60302$.

I'm hoping someone can explain why $r=(d\times b)^{0.5}$ is also $0.60302$.

Answer 1

$b = r \; \text{SD}_y / \text{SD}_x$ and $d = r \; \text{SD}_x / \text{SD}_y$, so $b\times d = r^2$.

Many statistics textbooks would touch on this; I like Freedman et al., Statistics. See also here and this wikipedia article.

Answer 2

Have a look at Thirteen Ways to Look at the Correlation Coefficient - and especially ways 3, 4, 5 will be of most interest for you.