how to explain $X$ and $Z$ Not relevant, how to prove it?

Question

Let the random variables $X$ and $Y$ be independent,$X$ follows the exponential distribution with parameter $1$, and the probability distribution of $Y$ is

$$\mathcal{P}\{Y=-1\}=p,\;\mathcal{P}\{Y=1\}=1-p;\;Z=XY$$

(1) Find the probability density of $Z$.

(2) $X$ and $Z$ are independent of each other?

This topic is a graduate entry topic. I know the probability density function of $X$.

$$f(x)=\begin{cases}e^{-x}, & \text{$x$>0} \\ 0, & \text{$x$≤0 } \end{cases}$$

Some people say its distribution function is

$$F(x)=\begin{cases}1-e^{-x}, & \text{$x$>0} \\ 0, & \text{$x$≤0 } \end{cases}$$ probability density of $Z$: $$f(z)=\begin{cases}pe^{z}, & \text{$z$≤0} \\ (1-p)e^{-z}, & \text{$z$>0 } \end{cases}$$

I want to know how to explain $X$ and $Z$ Independent?or not. how to prove it?Begging for the answer,Ask God for help.

Answer 1

Suppose towards a contradiction that $Z, X$ are independent. Then it follows that $|Z|$ and $|X|$ are independent (functions of independent random variables are independent). However, $|Z|=|X|$. Hence $|X|$ is independent of itself which is a contradiction.