Say $X,Y$ are i.i.d and $X,Y∼N(0,1)$. I would like to find joint pdf conditional on $X+Y>0$. More specifically $f_{XY}(x,x+y>0)$. I want to use random variable transform formula: $f_{U,Z}(u,z)=f_{XY}(g^{-1}_1(u,z),g^{-1}_2(u,z))\cdot |J(u,z)|$
I know that to calculate $f_{XY}(x,y)$ I can set $U=X$ and $Z=X+Y$ and proceed as usual. But what should be $Z$ for $X+Y>0$ ?
In general if $Z$ is a random variable with density $f_Z$ and $A$ is an event with $P(A)>0$ we have that $$ f_{Z\mid A}(z)=\frac{f_{Z}(z)}{P(A)}I(z\in A) $$ where $I$ is the indicator function since for any event $B$ $$ P(Z\in B\mid A)=\frac{P(Z\in A\cap B)}{P(A)}=\frac{1}{P(A)}\int_{B} f_{Z}(z)I(z\in A)\, dz=\int_{B} f_{Z\mid A}(z)\, dz. $$ Now returning to your question put $A=(X+Y>0)$. Then $$ f_{X,Y\mid A}(x,y)=\frac{f_{X,Y}(x,y)}{P(A)}=\frac{f_{X}(x)^2}{P(A)}=\frac{1}{P(A)}\frac{1}{2\pi}e^{-(x^2+y^2)/2};\quad (x.y)\in A $$ where we used the i.i.d assumption. To find $P(A)$ you can use the fact that $X+Y\sim N(0,2)$ where the second parameter is variance.