# Distribution of random variables

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.
-
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
@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
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

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.