Let (X,Y) be a bivariate normal and Z follow Bernoulli distribution and be independent of (X,y). Its mean $(X,Y)\sim N(\mu,\sigma I)$ and $Z \sim Ber(p)$. How can I find the joint distribution of them? And what are the MLE of mean and covariance of data?
In case of $(X,Y)$ be a bivariate normal I have some result:
$\hat{\mu}_1=\frac{1}{n}\sum_{i=1}^{n} x_{1i}$
$\hat{\mu}_2=\frac{1}{n}\sum_{i=1}^{n} x_{2i}$
$\hat{\sigma}_{11}=\frac{1}{n}\sum_{i=1}^{n} (x_{1i}-\hat{\mu_1})^2$
$\hat{\sigma}_{22}=\frac{1}{n}\sum_{i=1}^{n} (x_{2i}-\hat{\mu_2})^2$
and $\hat{\sigma}_{12}$ is the maximizer of 3-degree equation.
I want to add $Z\sim Ber(p)$ to them and find the MLE but I can't.