Help with independent random variables and correlation

[ychu066]

1 Let X be a normal variable with mean 0 and variance 1. Let Y = ZX
where Z and X are independent and Pr(Z = +1) = Pr(Z = -1) =1/2.

a Show that Y and Z are independent.
b Show that Y is also normal with mean 0 and variance 1.
c Show that X and Y are uncorrelated but dependent.
d Can you write down the joint density of X and Y ? Explain your
answer.

Note that this example exhibits two random variables which are un-
correlated, normally distributed, but not independent (and necessarily
not jointly normally distributed).

Please help me with the bold questions...

 

[fresh_42]

a.)
\begin{align*}
Pr(Z=z \wedge Y=y) &= Pr(Z=z \wedge zX=y)= Pr(Z=z \wedge X=y/z)\\
&=Pr(Z=z)\cdot Pr(X=y/z) = Pr(Z=z)\cdot Pr(zX=y)=Pr(Z=z)\cdot Pr(Y=y)
\end{align*}
We used that $X$ and $Z$ are independent, and that $Pr(Y)=Pr(zX)$ for any fixed $z\in\{\,-1,1\,\}$.

c.)
\begin{align*}
E(Y)=\int_\mathbb{R} y Pr(y)d\lambda_Y &= \sum_{z=\pm 1} \int_\mathbb{R} zx Pr(zx)d\lambda_X\\
&= -\dfrac{1}{2}\int_\mathbb{R} xd\lambda_X + \dfrac{1}{2}\int_\mathbb{R} xd\lambda_X = 0 = E(X)
\end{align*}
since $Z$ and $X$ are independent. And
\begin{align*}
E(XY)=\int_\mathbb{R} xy\,f(xy)d\lambda_{XY} = -\int_\mathbb{R} x^2\,f(-x^2)d\lambda_{X} + \int_\mathbb{R} x^2\,f(x^2)d\lambda_{X}=0
\end{align*}
since $X$ is symmetrically distributed at $x=E(X)=0$.
\begin{align*}Pr(X=x\wedge Y=y)&=Pr(X=x\wedge Y=zx)=Pr(X=x\wedge X=y/z=x)\\&=Pr(X=x)=Pr(Y=y)\end{align*}
and $Pr(X=x)Pr(Y=y)=Pr(X=x)^2 \neq Pr(X=x) = Pr(X=x\wedge Y=y)$