Suppose $X_1$ and $X_2$ are $N(0,1)$ random variables such that $X_2=-X_1$ if $|X_1|<1$ and $X_2=X_1$ otherwise. Obtain the cumulative distribution function of $X_1+X_2$ represented in the form(s) of cumulative distribution function of a standard normal variable.
I get $P(X_1+X_2<z)=\Phi (z/2)$ if $|z|>2$. But what about the case $|z|<2$? I would like to get convinced that my working is right. What I think will happen is, if $-2<z<2$ then $P(X_1+X_2<z)=P(X_1+X_2<-2)+P(-2<X_1+X_2<z)=\Phi (-1)+P(-2<0<z)=\Phi (-1)+0.5$
This is because $0$ is either less than or greater than $z$ which is a real number. So $P(-2<0<z)=P(0<z)=0.5$.
Is the reasoning correct? Thanks in advance.