# Finding joint distribution of $(X + Y,X^2 + Y^2)$ where $X,Y$ are independent standard normal variables

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 $$Z \sim \chi^2_2$$.

• 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.
– Ben
Nov 4, 2019 at 7:03

## 1 Answer

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 $$\frac{\partial (\xi, \eta)}{\partial (w, z)}$$.