# 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?

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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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