Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $X_i: \Omega \rightarrow\mathbb{R}$ random variables for $i=1,2$.
$X_1$ and $X_2$ are jointly Gaussian distributed with joint density on $\mathbb R^2$:
$$p(x,y)= \frac{\exp\left[-\frac{1}{2(1-\rho^2)}(\frac{(x-\mu_1)}{\sigma_1}-\frac{(y-\mu_2)}{\sigma_2})^2\right]}{2 \pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}}$$
Now I need to calculate $\mathbb{E}(X_i)$ and Var($X_i)$.
So far I only have that:
$$\mathbb{E}(X_i)=\int_{\Omega}^{}X_id\mathbb{P}= \int_{\mathbb{R}}^{}xp_{X_i}dx$$
but $p_{X_i}(x)=\int_{\mathbb{R}}^{}p(x,y)dy$ and I don´t know how one would compute this integral or if its even possible.
Can someone help?