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I have $X \sim N(0,1)$ and discrete $Y$ independent of $X$ with $P(Y=1)=P(Y=-1) = 1/2$. $Z$ defined as $Z = XY$. Show $X$ and $Z$ are uncorrelated and not independent.

I have shown that they are uncorrelated by having $\mathrm{cov}(X,Z) = 0$ with $E(X)=E(Z)=0$.

How can I show that they are independent?

Thank you

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Note that $|X|=|Z|$, while if $X$ and $Z$ were independent normal random variables then we would have $\mathbb{P}(|X|=|Z|)=0$.

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    I am sorry - could you please expand why in case of independence P(|X|=|Z|)=0 ? – 2017-02-14
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    If $X$ and $Z$ are independent then they have a joint pdf, and the set $\{(x,z):|x|=|z|\}$ has measure zero in $\mathbb{R}^2$ so $\mathbb{P}(|X|=|Z|)=0$. – 2017-02-14