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If $X$ is independent of $Y$ and $Y$ is independent of $Z$, is $X$ independent of $Z$? In other words, is independence transitive?

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No. in the extreme case, take for exammple $X=Z$.

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You may also think about real life example:

  1. $X$ - You find a lottery ticket on the street
  2. $Y$ - You flip a coin and get a head
  3. $Z$ - You win the lottery.

$X$, $Y$ and $Y$, $Z$ are independent, but lottery ticket for sure help you to win some extra cash.

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