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Please provide an example for random variables $X, X',Y ,Y'$ such that:

  1. $X,X'$ have same distributions
  2. $X+Y$ and $X'+Y'$ have same distributions but $Y,Y'$ haven't same distributions.
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what else do we have? please make sure you have your question ready before hitting the "post" button. –  jay-sun Jan 14 '13 at 18:37
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@yalda: Welcome to MSE! Is there an actual question as it is not clear what you are asking? Why are you bolding the entire statement? Regards –  Amzoti Jan 14 '13 at 18:43
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Why are you flouting the advice given to you on your other questions about the proper way to ask questions on MSE? –  Did Jan 14 '13 at 20:02
    
Hint: take $Y=-2X$. –  Greg Martin Jan 14 '13 at 21:04

1 Answer 1

up vote 5 down vote accepted

Toss a fair coin once. Let $X$ be the number of heads. Let $X'$ be the number of heads (not a typo). Obviously $X$ and $X'$ have the same distribution.

Let $Y$ be $0$ with probability $1$, and let $Y'$ be the number of tails minus the number of heads.

Then $X+Y$ is the number of heads, and $X'+Y'$ is the number of tails. So $X+Y$ and $X'+Y'$ have the same distribution.

But it is clear that $Y$ and $Y'$ do not have the same distribution.

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thanks for your answer –  yalda Jan 15 '13 at 5:24
    
Andre do you have another example? –  user58635 Jan 17 '13 at 16:25

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