How to define a distribution such that draws from it correlate with a draw from another pre-specified distribution? How do I define the distribution of a random variable $Y$ such that a draw from $Y$ has correlation $\rho$ with $x_1$, where $x_1$ is a single draw from a distribution with cumulative distribution function $F_{X}(x)$?  
 A: You can define it in terms of a data generating mechanism. For example, if $X \sim F_{X}$ and 
$$ Y = \rho X + \sqrt{1 - \rho^{2}} Z $$ 
where $Z \sim F_{X}$ and is independent of $X$, then, 
$$ {\rm cov}(X,Y) = {\rm cov}(X, \rho X) = \rho \cdot {\rm var}(X)$$ 
Also note that ${\rm var}(Y) = {\rm var}(X)$ since $Z$ has the same distribution as $X$. Therefore, 
$$ {\rm cor}(X,Y) = \frac{ {\rm cov}(X,Y) }{ \sqrt{ {\rm var}(X)^{2} } } = \rho $$ 
So if you can generate data from $F_{X}$, you can generate a variate, $Y$, that has a specified correlation $(\rho)$ with $X$. Note, however, that the marginal distribution of $Y$ will only be $F_{X}$ in the special case where $F_{X}$ is the normal distribution (or some other additive distribution). This is due to the fact that sums of normally distributed variables are normal; that is not a general property of distributions. In the general case, you will have to calculate the distribution of $Y$ by calculating the (appropriately scaled) convolution of the density corresponding to $F_{X}$ with itself. 
