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

Question

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?

Answer 1

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 $W$.