For mixture of multivariate Bernoulli distribution we have that,
$p(x|\mu,\pi) =\Sigma_{k=1}^{K}\pi_kp(x|\mu_k)$ where $p(x|\mu_k) = \prod_{i=1}^{D}\mu_{ki}^{x_i}(1-\mu_{ki})^{1-x_i}$
I read it from the book that
$E[x] = \Sigma_{i=1}^{K}\pi_k\mu_k$ $cov[x] = \Sigma_{k=1}^{K}\pi_k(\Sigma_k+\mu_k\mu_k^T) - E[x]E[x]^T$
The mean is trivial to prove, however I can't find proof for the covariance and I don't know how to prove it.
Can anyone help?