Normal Random Variable transformation distribute as N(0,1)

Question

We know that a random variable Y $\sim N(0,1)$

We define the random variable $Z= Y \mathbb 1_{|Y|\leq a} - Y \mathbb 1_{|Y|>a}$

I want to proove that Z $\sim N(0,1)$, and I know that I've to do the transformation, but I don't know how to do it because of the absolute value in the indicator function. How can I do that transformation?

Thank you very much!

Answer 1

$$P(Z \leq z)=P(|Y| \leq a, Y \leq z)+P(|Y| > a, -Y \leq z)=P(|Y| \leq a, Y \leq z)+P(|Y| > a, Y \leq z)$$ $$=P(Y \leq z)$$ where the second equality stems from the fact that $Y$ and $-Y$ have the same distribution.