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Give examples of (not independent) random variables $X$ and $Y$, both of which are uniform in the interval $[0, 1]$ and such that

  1. $\mathbb{P}(X + Y = 1) = 1$
  2. $X + Y$ is uniform in the interval $[0, 2]$.
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For 1. take $Y=1-X$. Prove that $Y$ is uniformly distributed.

For 2. Take $Y= \alpha X$. Prove that $X+Y = (1+\alpha) X$ is uniformly distributed on $[0,1+\alpha]$. For which $\alpha$ will $Y$ be uniform on the unit interval?

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I don't know if that constitutes an example? – Olivia Irving Jan 17 '13 at 17:21

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