In my quest to understand all things Spearman, consider the following problem:

Given random variable $x$ with known variance, $\sigma^2$, and $p \in [-1,1]$, one can construct a random variable $y$ such that the Pearson correlation coefficient of the variables $x$ and $y$ is exactly $p$. (Let $y = p x + \sqrt{(1-p^2)} z,$ where $z$ is a random variable independent of $x$ with variance $\sigma^2$.)

I am wondering if there is an analogue for the Spearman Rank Correlation Coefficient. I am defining the *population* Spearman Correlation Coefficient of the jointly distributed $(x,y)$ as
$$E[sign((x_i - x_j)(y_i - y_j))].$$
(this is the subject of another of my questions here on M.O.)

It seems like I would need to know more than just the variance of $x$ to construct $y$, perhaps the entire CDF.