Covariance of a mixture of Gaussians I have seen this question asked, but in a strange way that I do not think is equivalent. If someone can show that the formulations are identical, I would be grateful.
Suppose with probability $p$, one draws points from a Gaussian distribution $N(0,\hat\Sigma_1)$, and with probability $1-p$ from a distribution $N(0,\hat\Sigma_2)$, where $\hat\Sigma_1$ and $\hat\Sigma_2$ are both $p,p$ matrices. After $n$ such draws, one can compute the covariance matrix of your points $\Sigma$.
Can we have an explicit formula for $\Sigma$?

The distribution of the variable is
$$
X = 1_{U<p} X_1 + 1_{U\ge p} X_2
\sim 1_{U<p} N(0, \hat\Sigma_1) + 1_{U\ge p} N(0, \hat\Sigma_2) 
$$ 
(with $U\sim U[0,1]$ is independent of the rest)
which as expected value $0$ and second moment
\begin{align}
E[X_iX_j] 
&= E[(1_{U<p} X_{i,1} + 1_{U\ge p} X_{i,2})(1_{U<p} X_{j,1} + 1_{U\ge p} X_{j,2})]
\\&= E[1_{U<p} X_{i,1}  X_{j,1}] + E[1_{U\ge p} X_{j,2}  X_{j,2}]
\end{align}
now if you expand everything and use the independence of $U$ and $X_{1,2}$ you should get the matrix
$$
p\hat\Sigma_1 + (1-p)\hat\Sigma_2
$$
