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Suppose $X_1$ is a standard normal random variable. Define $$X_2=\begin{cases} -X_1, &\text{if} \,\, |X_1|<1 \\ \,\,\,\,X_1, & \text{otherwise}\end{cases}$$ Obtain the cumulative distribution function of $X_1+X_2$ in terms of the cumulative distribution function of a standard normal random variable.

Trial: Define $Y=X_1+X_2$. So we have $$Y=\begin{cases} \,\,0, &\text{if} \,\, |X_1|<1 \\ 2X_1, & \text{otherwise}\end{cases}$$Then I am stuck. Please help. Thanks in advance.

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what you have written is pretty correct so far. what you need to do. so the CDF is calculated as

$P(Y\leq y)$. You have to work this out for the 3 regions $X\leq -1$, $-1<X<1$,and $X\geq 1$.

The first one is related to that a normal distribution right? because the lower tail has not really changed, you can substitute in $2X$ where $Y$ is.

the second one should be a constant (why? what is it?)

The last one is probably the hardest, but you can do $P(Y\leq y) = 1-P(Y>y)$ which is similiar to the first region.

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It seems that $F_Y(y)=F_X(u(y))$ with $u(y)=\frac12y$ if $|y|\geqslant2$, $u(y)=-1$ if $-2\leqslant y\lt0$ and $u(y)=+1$ if $0\leqslant y\lt2$.

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