Distribution of $\frac{X+Y}{X-Y}$

Question

It is given that X and Y are independent Gaussian random variables with mean 0 and variance $\mu$.

The distribution of $\frac{X+Y}{X-Y}$ is asked.

If
Z = X+Y
W = X-Y

Then Z and W are also independent ( because cov(Z,W) = 0 ).
How to proceed further?

Answer 1

$$P\left(\frac{X+Y}{X-Y} <t\right) =\int_{-\infty}^t \frac{1}{\pi (1+u^2)} du =\frac{\arctan t}{\pi} +\frac{1}{2}$$

Answer 2

The distribution of $X+Y$ is $\mathcal N(0,2\mu)$ and the distribution of $X-Y$ is the same $\mathcal N(0,2\mu)$. As you pointed out, $X+Y$ and $X-Y$ are independent, so we have a ratio of two independent normal random variables.

When $U$ and $V$ are two independent normally distributed random variables with expected value $0$ and variance $1$, then the ratio $U/V$ has the standard Cauchy distribution.

We have that $$ \frac{X+Y}{X-Y}=\frac{(2\mu)^{-1/2}(X+Y)}{(2\mu)^{-1/2}(X-Y)}. $$ Hence, the distribution of the ratio is the standard Cauchy distribution.