# Marginal distributions of two linear transformations of two correlated Gaussian (Normal) distributions

Considering this entry the distribution of the sum of non i.i.d. gaussian variates is also gaussian.

\begin{align*} V = aX + bY &\sim N(a\mu_X + b\mu_Y,\; a^2\sigma_X^2 + b^2\sigma_Y^2 + 2ab\sigma_{X,Y}) \end{align*}

What if we have another transformation like W as below:

\begin{align*} W = cX + dY &\sim N(c\mu_X + d\mu_Y,\; c^2\sigma_X^2 + d^2\sigma_Y^2 + 2cd\sigma_{X,Y}) \end{align*}

Knowing that X and Y are identically distributed and correlated Gaussian r.v.s what are the marginal distributions of V and W?

Besides, how to get the correlation coefficient of V and W?

You have already written out the marginal distributions for $$V$$ and $$W$$ above -- they are normal with means and variances as you expressed. To find their correlation, we can write:
$$\text{cov}(V,W) = \text{cov}(aX+bY,cX+dY)$$
which in turn can be decomposed as $$\text{cov}(aX,cX) + \text{cov}(aX,dY) + \text{cov}(bY,cX) + \text{cov}(bY,dY).$$ Notice that $$\text{cov}(aX,cX) = ac \cdot \text{var}(X)$$ and $$\text{cov}(bY,dY) = bd \cdot \text{var}(Y)$$. We also know that $$\text{cov}(aX,dY) = ad \cdot \text{cov}(X,Y)$$ and $$\text{cov}(bY,cX) = bc \cdot \text{cov}(X,Y)$$. So, you can compute $$\text{cov}(V,W)$$ by just plugging in these values, which you already know from the joint distribution of $$X$$ and $$Y$$.
Finally, to get the correlation coefficient, you can just divide your value of $$\text{cov}(V,W)$$ by the square root of the variance of $$V$$ and the square root of the variance of $$W$$.