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?

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

Your Answer

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

Browse other questions tagged or ask your own question.