Solved – 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?

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

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

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