X, Y are independent standard normal random variables, what is the distribution of $ \frac{X}{X+Y} $
Could anyone help me with this? Thanks.
I have worked the problem by multivariable transformation:
Let $Z=\frac{X}{X+Y} , W=X$
Consider transformation $(X,Y)\longrightarrow(Z,W)$
Then $X(Z,W)=W , Y(Z,W)=\frac{W(1-Z)}{Z}$ defines the inverse transformation.
The Jacobian is $J(Z,W)=\frac{w}{z^{2}} $
So $f_{Z,W}(z,w)=f_{X,Y}(w,\frac{w(1-z)}{z})\cdot\mid\frac{w}{z^{2}}\mid$
As X and Y are independent. Then the marginal pdf of Z is $f_{Z}(z)=\intop_{0}^{\infty}\frac{w}{z^{2}}\cdot f_{X}(w)\cdot f_{Y}(\frac{w(1-z)}{z})dw+\intop_{-\infty}^{0}-\frac{w}{z^{2}}\cdot f_{X}(w)\cdot f_{Y}(\frac{w(1-z)}{z})dw$ After calculation we get $f_{Z}(z)=\frac{1}{\pi\cdot\frac{1}{2}\cdot(1+(\frac{z-\frac{1}{2}}{\frac{1}{2}})^{2})}$
Hence $Z\sim \mathrm{Cauchy}(\frac{1}{2},\frac{1}{2}).$