Joint distribution of sum of independent normals Suppose we have three independent normally distributed random variables 
$$ X_0 \sim \mathcal{N}(\mu_0, \sigma_0^2), $$
$$ X_1 \sim \mathcal{N}(\mu_1, \sigma_1^2), $$ 
$$ X_2 \sim \mathcal{N}(\mu_2, \sigma_2^2).$$
Now, define two new random variables $Y_0 = X_0+X_1$ and $Y_1 = X_1+X_2$. 
Let $\vec{Y} = [Y_0 \;\;\; Y_1]^T$
What can we say about the distribution of $\vec{Y}$? Obviously, $Y_0$ and $Y_1$ are not independent. If they were, then $\vec{Y}$ would have been a multivariate normal variable. Any ideas? 
 A: 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 
$$ f(y_0,y_1) = N\left(\vec{\mu} = \begin{bmatrix}
           \mu_0+\mu_1 \\
           \mu_1+\mu_2 \\         
         \end{bmatrix},  \Sigma = \begin{bmatrix}
           \sigma^2_0+\sigma^2_1 &\sigma_1^2 \\
           \sigma_1^2 & \sigma^2_1+\sigma^2_2 \\         
         \end{bmatrix}  \right).
$$
