# Covariance matrix of integral of multivariate normal distribution

If $$t = [t_0, t_1, \dots, t_{N-1}] \in \mathbb{R}^N$$ with $$t_i \sim N(\mu_i, \sigma_i^2)$$ and its covariance matrix $$C \in \mathbb{R}^{N \times N}$$ where $$C_{ij} = Cov(t_i, t_j)$$ is given

If I define a multi bernullian distribution $$y = t > 0$$ where $$y_i = t_i > 0$$ (1 if true, zero if false)

$$E[y] = 1 - \frac{1}{2}(1 +erf(-\frac{E[t]}{\sqrt{2}Var[t]}))$$

Where $$Var[t] = [C_{00}, C_{11}, \dots, C_{(N-1)(N-1)}]$$

How can I compute the covariance matrix of $$y$$? Did I compute $$E[y]$$ correctly?

• The calculation comes down to the bivariate case, requiring integrating the bivariate density over quadrants. See jstor.org/stable/2006276?seq=1, for instance. – whuber Jul 22 at 15:24