Converting a normal RV to Standard Normal RV and distribution of a 3-case RV

Question

Let $ X \sim N(0,2)$ and $Y=X^2$

And let $$ Z= \begin{cases} 1 & X>0 \\ 0 & X =0 \\ -1 & X<0 \end{cases} $$

And I would like to know the relationships between Y and Z (are they dependent, coordinated etc.

Iv'e a few questions regarding that:

1. Could it help to convert X to the standard normal form? When would one do that? if I should, how can I do that?

2. Does Z consider to be continuous or discrete? because I learned that discrete RV usually have this kind of "steps" graph, like the above Z, but on the other hand, it doesn't look countable, so I am confused about it.

3. Can I simply calculate $E[Z]$ by integrating all 3 regions and summing it? (if continuous of course).

Thank you so much.

Answer 1

Since $Z$ can only take on three values, it is discrete. We can work out its marginal distribution as follows: \begin{align*} P(Z = 1) &= P(X > 0) = 1/2\\ P(Z = 0) &= P(X = 0) = 0\\ P(Z = -1) &= P(X < 0) = 1/2 \end{align*} using the fact that the $N(0,2)$ distribution is symmetric about zero and continuous. So we see that $Z$ is a Rademacher random variable. Now, the question is how does it relate to $Y = X^2$? To answer this question, think about how $Z$ was constructed: it only contains information about the sign of $X$. In contrast, $Y$ tells us nothing about the sign of $X$. Hence, $Y$ and $Z$ are independent.