I'm just confused on how to set up and start this problem. I'm confident that once I start down the right path, I'll have little issue.
Let $p_1$ denote a bivariate normal distribution $N(0, 0, 1, 1, 0)$, and $p_2$ denote a bivariate normal distribution $N(0, 0, 1, 1, \rho)$. Suppose they have joint density:
$$p(x, y) = \frac{1}{2}p_1(x,y) + \frac{1}{2}p_2(x,y)$$.
I have to show that $X$ and $Y$ have marginally normal densities, but that their joint density is normal iff $\rho=0$.
I want to accomplish this through the use of matrix notation, instead of via the pdf.
The problem I'm having is that the problem doesn't say $p_1$ and $p_2$ are independent, and I believe they aren't since, for example:
$$Cov(X_{p_1},X_{p_2})=\mathbb{E}(X_{p_1}X_{p_2})+\mathbb{E}(X_{p_1})\mathbb{E}(X_{p_2})=\mathbb{E}(X^2)+0=1$$.
And the same for $Y$. Even though they're not independent, I guess I could still reduce them to their marginal distributions:
$$ p_1(X) \sim N\left(\begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} \mu_{X_1}\\\mu_{Y_1} \end{pmatrix}, \begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \right) $$
And so on for $p_2(X)$. Then, I would find $p(X)$ by:
$$\mathbb{E}(p(X))= \frac{1}{2}\mathbb{E}(p_1(X))+\frac{1}{2}\mathbb{E}(p_2(X))$$
and
$$Var(p(X))=\left(\frac{1}{2}\right)^2 Var(p_1(X)) + \left(\frac{1}{2}\right)^2 Var(p_2(X)) + 2 Cov(X,X)$$
And similarly for Y.
Even if they were independent, though, I wouldn't know exactly how to set up the joint density part of the problem in matrix terms. For example, would it be (using $\mathbf{W}=\begin{pmatrix} 1/2 & 1/2 \end{pmatrix}$):
$$\mathbf{p}\sim N\left( \mathbf{W} \boldsymbol{\mu}_{p_1} + \mathbf{W} \boldsymbol{\mu}_{p_2}, \mathbf{W} \boldsymbol{\Sigma}_{p_1} \mathbf{W}^{T} + \mathbf{W} \boldsymbol{\Sigma}_{p_2} \mathbf{W}^{T} \right)$$
I would appreciate some insight into how to set up this problem.