Suppose $X_1$ is a standard normal variable. Define, $$X_2=\begin{cases}-X_1, & \text{if } |X_1|<1,\\X_1, & \text{otherwise}\end{cases}$$
Show that $X_2$ is also a standard normal random variable.
My approach:
Let,$$F(X_2)=P[X_2\leq x_2]\\=P[X_2\leq x_2\mid|X_1|\leq 1].P[|X_1|< 1]+P[X_2\leq x_2\mid|X_1|\geq 1].P[|X_1|\geq 1]\\=P[-X_1\leq x_2].P[|X_1|<1]+P[X_1<x_2].P[|X_1|>1]$$
But I cannot compute the probabilities.May be,my approach is not right.