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 $F_{X}(x)$?


1 Answer 1


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.

  • 2
    $\begingroup$ +1 Very nice answer. Nitpick: in the last line you need to convolve scaled versions of $F_X$. $\endgroup$
    – whuber
    Jul 22, 2011 at 16:53
  • $\begingroup$ Thanks so much, Macro. Just to clarify something -- you mean in your last paragraph that you would need to convolve the rho*X with the sqrt(1 - rho^2)*X? (sorry, I couldn't get any formatting, even HTML to work in this particular comment) $\endgroup$
    – OctaviaQ
    Jul 22, 2011 at 23:27
  • 1
    $\begingroup$ Convolve the densities corresponding to the distributions of $\rho X$ with the distribution of $\sqrt{1 - \rho^{2}} X$. This is a result of the general fact that the density of the sum of two continuous random variables is the convolution of the two densities. $\endgroup$
    – Macro
    Jul 23, 2011 at 1:51
  • 1
    $\begingroup$ A long time but...ideas of how to do this, also enforcing the marginal distribution of Y? $\endgroup$ Mar 22, 2015 at 21:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.