# Covariance of two truncated random variables

Assume that $X=(X_1,X_2)^T$ is a random vector that is multivariate normal with mean vector $\mu$ and variance-covariance matrix $\Sigma$. I.e. we have $$X \sim N_2(\mu, \Sigma).$$

I am interested in the truncated version of this normal distribution and assume that $X_1$ and $X_2$ are smaller than some threshold $c$. Hence I am searching for $X^*=(X_1^*,X_2^*)^T$ that is jointly trucnated normal

$$X^*~\sim TN_2(\mu, \Sigma,c)$$ While there are formulas for the expectation and the variance of the univariate truncated normal I do not know how to compute the covariance of $X_1^*$ and $X_2^*$: $\operatorname{Cov}(X_1^*,X_2^*)= \operatorname{Cov}(X_1,X_2\mid X_1<c,X_2<c)$.

• This is not going to be pretty. First of all, recall that a shortcut formula for the covariance is $$\text{Cov}(X_1^{*}, X_2^{*}) = \mathbb{E}[X_1^{*}X_2^{*}] - \mathbb{E}[X_1^{*}]\mathbb{E}[X_2^{*}]$$ so, you would have to find $\mathbb{E}[X_1^{*}X_2^{*}]$ using the joint density, which I imagine would be a disgusting integral that will probably have to be numerically approximated. – Clarinetist Feb 27 '18 at 12:50
• @Clarinetist you should never forget that marginal expectations here are conditioned on both $X_1$ and $X_2$ - one should keep this in mind – Denis Korzhenkov Feb 27 '18 at 12:52