Joint Distribution of Two Dependent Variables having the Marginals

Question

From two Independent Normal Random Variables: $X \sim N(\mu_1,\sigma) $ and $Y\sim N(\mu_1,\sigma)$, I created two DEPENDENT random variables $Z$ and $W$:

$Z= X - Y$

$W= X - g(Y)$ where $g(\cdot)$ monotonic transformation

I would like to find the joint distribution $f_{Z,W}$.

(I have read that I might need the Copula, but I believe that having the marginals PDFs and the equations of $Z$, $W$ and $g(\cdot)$ the Copula should be specified somehow and not chosen randomly)

Thanks in Advance!

Answer 1

From two Independent Normal Random Variables: $X\sim N(\mu_1,\sigma)$ and $Y\sim N(\mu_1,\sigma)$ , I created two DEPENDENT random variables Z and W:

Be careful because for a particular choice of $g$ the random variables you created are INDEPENDENT

Counterexample:

Set $g(Y)=-Y$ thus you have

$$Z=X-Y$$

$$W=X+Y$$

and obviously

$$Cov(Z,W)=\mathbb{E}[X^2]-\mathbb{E}[Y^2]=0$$

in a gaussian model, this is equivalent as independence.