Solved – Joint distribution of sum of independent normals

Not entirely clear to me from reading the comments if the OP has solved this but there is no answer so I will write one.

The distribution of each $Y_i$ will be normal with given means and variances:

$mu_0+mu_1$ and $sigma_0^2+sigma^2_1$ for $Y_0$ and

$mu_1+mu_2$ and $sigma_1^2+sigma^2_2$ for $Y_1$. Now finally we need to determine if there is a correlation between $Y_0$ and $Y_1$. To do this we can calculate

$$mathbb{C}ov(Y_0,Y_1)=mathbb{C}ov(X_0+X_1,X_1+X_2) =mathbb{C}ov(X_1,X_1) =mathbb{V}ar(X_1) =sigma_1^2. $$ Now you can turn this into a correlation by dividing by the square roots of the variances

$$rho = frac{sigma_1^2}{sqrt{(sigma_0^2+sigma^2_1)(sigma_1^2+sigma^2_2)} }.$$

Now we know that the sum of two normal random variables is normally distributed so that both $Y_0$ and $Y_1$ have normal distributions with the stated means and variances and the correlation is given by $rho$ above. So the joint density of $Y_0, Y_1$ is

$$ f(y_0,y_1) = Nleft(vec{mu} = begin{bmatrix} mu_0+mu_1 \ mu_1+mu_2 \ end{bmatrix}, Sigma = begin{bmatrix} sigma^2_0+sigma^2_1 &sigma_1^2 \ sigma_1^2 & sigma^2_1+sigma^2_2 \ end{bmatrix} right). $$