I'm facing difficulty in trying to prove the following:
Let $X_1$, $X_2$ and $\xi$ be random variables such that $X_1 \sim \mathrm N(0,1)$ and $X_2 = |X_1| \!\cdot\! \xi$, where $\xi = \{-1, 1\}$ with $1/2$ probability. Also, $\xi$ is independent of $X_1$. Prove that $X_2 \sim \mathrm N(0, 1)$, where $\mathrm N(0, 1)$ is a standard normal distribution with mean $0$ and variance $1$.
I managed to compute $$P(X_2 \leq x) = \tfrac12 (1+P(|X_1| \leq x)).$$ But how to reduce the right hand side to $P(X_1 \leq x)$?
Thanks in advance.