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Take three random variables $X$, $Y$, $Z$ s.t. $E[X]>0$, $E[Y|X]=0$, $Z = X+Y$.

What can I say about $E[x| x> k]$ vs. $E[z| z>k]$ where $k>0$? Intuitively, the latter should be bigger but I have failed to (dis)prove it. Any hint/reference?

user312267
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    Let $(X,Y)\in\{(1,-1/4),(1,1),(2,0),(10,0)\}$ with probabilities $1/10,4/10,1/4,1/4$ respectively. Verify this satisfies all your conditions, then consider $k=1.$ – whuber Feb 23 '21 at 13:50
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    Thanks a lot @whuber! Probabilities should be $4/10$, $1/10$, $1/4$, $1/4$ but all the rest checks out. I guess the take away is that noise can add mass in the center but take it away from tails. I wonder about the sufficient conditions so that noise adds mass in the right tail as well, but I will need to think about this more carefully – user312267 Feb 23 '21 at 14:37
  • Sorry about transposing the probabilities--I'm glad you understood my intention. – whuber Feb 23 '21 at 14:49

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