How would the covariance matrix $\Sigma$ for P(T,Y) be calculated from an equation of the form:
$X:=\mathcal{N}(0,1)$
$T:=X+\mathcal{N}(0,1)$
$Y:=X+T+\mathcal{N}(0,1)$
Edit: each $\mathcal{N}(0,I)$ is independent
Using the useful fact below, solving for the mean vector and diagonal values of $\Sigma$ of P(T,Y) is simple enough, however it isn't clear to me how one would solve for $\Sigma_{TY}$
Useful:
if $X \sim \mathcal{N}(\mu_X,\sigma_X^2)$, $Y \sim \mathcal{N}(\mu_Y,\sigma_Y^2)$, and $Z = X + Y$
then, $Z \sim \mathcal{N}(\mu_X + \mu_Y,\sigma_X^2 + \sigma_Y^2)$
Thank you for any responses