If a PDF is re-constructed as following $p(x)=\sum_{k=1}^{K}\pi_k p(x|k),$ where $p(x|k)$ has mean $\mu_k$ and covariance matrix $\Sigma_k$ , compute the mean and covariance of the new mixture distribution.
Finding the mean is trivial , so say that I've found the mean and denote it as $mean=\left[ \begin{array}{cl} mean_{1} \\ ... \\ mean_{D} \end{array}\right]$
I'm quite confused about the dimension of covariance matrix, say I have $D$ dimension for each observation $x=\left[ \begin{array}{cl} x_{1} \\ ... \\ x_{D} \end{array}\right]$, then dimension of covariance matris should be $D \times D,$ but what 's the concrete element in position $$ ?