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Given X ~ N(0,1) , Y ~ Ber(1/4)

Z =

 X      if Y = 0
-X      if Y = 1

How to approach this question if i want to prove that Z and X are dependent .

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  • $\begingroup$ Hint: how do the distributions of $X$ and $-X$ differ? $\endgroup$ – whuber Jul 1 at 14:22
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Two approaches

  • Consider $P(Z>0 | X>0)$, which would be 0.5 if they were independent.
  • For independent continuous variables $X$, $Z$, what is $P(X=Z)$?

The second approach is more powerful; it even works when $Y$ is Ber(1/2). If this is a teaching question I suspect the first one is what was intended.

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