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Let $X$ and $Y$ be jointly distributed as a multivariate normal with the following parameters:

$$ \mu_{XY} = \begin{bmatrix} 0 \\ 0.2 \end{bmatrix} \qquad \Sigma_{XY} = \begin{bmatrix} 1 & 0.05 \\ 0.05 & 0.16 \end{bmatrix} $$

Now assume a third random variable, $Z = f(Y)$, created with the intention of discretizing $Y$. Specifically, we have

$$ Z = \begin{cases} 1 &y < F^{-1}(0.2)\\ 0 &\mathrm{otherwise} \end{cases}, $$ where $F^{-1}$ is the quantile function for $Y$, which means $F^{-1}(0.2)\approx -0.136$.

So I assume $Z \sim Bernoulli(0.2)$, which gives it the same mean and variance as $Y$.

My question is: how should the covariance matrix between $X$, $Y$ and $Z$ look? This is what I can immediately fill in:

$$ \Sigma_{XYZ} = \begin{bmatrix} 1 & 0.05 & ? \\ 0.05 & 0.16 & ? \\ ? & ? & 0.16\\ \end{bmatrix} $$

Which means that I need to figure out the covariances involving $Z$. My first instinct was that $\sigma_{XZ} = \sigma_{XY} = 0.05$ and $\sigma_{YZ} = 1$, but some simulated data showed this is not true. I can't get an analytic result either.

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  • $\begingroup$ Yes, what I need in the end is a "true" covariance matrix, so I can do two things: 1) when I generate data from these three r.v., I want to compare the observed covariance matrix with the true one. 2) I want to compare linear regression coefficients generated from the data with some references, which I would retrieve from this "true" covariance matrix. $\endgroup$ – Waldir Leoncio Dec 31 '18 at 16:21
  • $\begingroup$ @JesperHybel, yes, right again! I got the categories flipped. I'll fix it right away. I had them right on paper and on code, though, so I'm still stuck. $\endgroup$ – Waldir Leoncio Dec 31 '18 at 16:38
  • $\begingroup$ The covariance $cov(Z,Y) = \mathbb E[ZY] - p \mathbb E[Y]$. And $\mathbb E[ZY] = \mathbb E[ZY\lvert Z=1] p(Z=1) + \mathbb E[ZY\lvert Z=0] p(Z=0)$ which reduce to $\mathbb E[Y\lvert Z=1] p(Z=1) = \mathbb E[Y \lvert Y < F_Y^{-1}(0.2)] p(Z=1)$. This is the expectation of a truncated normal link. No closed form solution, but can be expressed using cdf and pdf of normal. $\endgroup$ – Jesper Hybel Dec 31 '18 at 17:14
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You could perhaps use the following

$Cov(Z,Y) = \mathbb E[ZY] - Pr(Z=1) \mathbb E[Y]$

found simply by applying the definition of covariance. Focusing on the term $\mathbb E[ZY]$ it then follows by total law of expectation that

$$ \mathbb E[ZY] = \mathbb E[ZY\lvert Z=1]Pr(Z=1) + \underbrace{\mathbb E[ZY\lvert Z=0]}_{=0}Pr(Z=0)$$

where one summand is seen to be $0$ such that the indentity becomes

$$ \mathbb E[ZY] = \mathbb E[ZY\lvert Z=1]Pr(Z=1) \\ = \mathbb E[Y\lvert Y < F_Y^{-1}(0.2)]Pr(Z=1) $$

This is the expectation of a truncated normal. No closed form solution, but can be expressed using cdf and pdf of normal.

Same logic should be applicable to $Cov(Z,X)$ to get

$Cov(Z,X) = \mathbb E[X\lvert Y < F_Y^{-1}(0.2)]Pr(Z=1) - Pr(Z=1) \mathbb E[X]$

and then you can look here link for an expression of the expectation $\mathbb E[X\lvert Y < F_Y^{-1}(0.2)]$ which is the only part assumed not given in the problem.

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    $\begingroup$ Thank you for taking the time to write this. The Math checks out, but I am still working it out in practice. Happy 2019! $\endgroup$ – Waldir Leoncio Jan 1 at 10:05

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