Find joint distribution of $W = X + Y$ and $Z = X^2 + Y^2$ where $X,Y \stackrel{\text{i.i.d}}\sim\mathcal{N}(0,1)$.

I am trying to do this by the change of variable method.

So first I need to get $X,Y$ in terms of $Z,W$. Doing this we get:

Case 1: $$X = \frac{W - \sqrt{2Z - W^2}}{2}\quad,\quad Y = \frac{W + \sqrt{2Z - W^2}}{2}$$

Case 2: $$Y = \frac{W - \sqrt{2Z - W^2}}{2}\quad,\quad X = \frac{W + \sqrt{2Z - W^2}}{2}$$

So the two are similar.

Now I can use these to compute the Jacobian then make the substitution. My question is: Is there a better way to do this or a well known theorem/result I can use here?

Curious since $W \sim \mathcal{N}(0,2)$ and $Z \sim \chi^2_2$.

  • $\begingroup$ This is a special case of the famous Cochran theorem, for $n=2$ data points. I recommend you look up this famous theorem as a part of looking at this problem. $\endgroup$ – Ben Nov 4 '19 at 7:03

Consider first the joint distribution of $\xi = W/2$ and $\eta = X^2 + Y^2 - 2\xi^2 = Z - W^2/2$, which is well-known (think of the joint distribution of sample mean $\bar{X}$ and sample variance $S^2$ for a normally-distributed sample).

You can then get the joint distribution of $(W, Z)$ by writing out the joint density of $(\xi, \eta)$ and computing the Jacobian $\frac{\partial (\xi, \eta)}{\partial (w, z)}$.

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