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Sep 12, 2011 at 13:33 vote accept Apeirohedron
Sep 12, 2011 at 12:49 comment added fabee I updated the answer. If you know $\sigma_x$ and $\sigma_z$ then the above is your expectation (please double-check them before you use them for anything important).
Sep 12, 2011 at 8:14 history edited fabee CC BY-SA 3.0
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Sep 12, 2011 at 7:26 comment added Apeirohedron What happens if the distributions are not independent? Sorry, I'm thinking through this again and I think I might have misunderstood. In fact, $Z$ and $X$ are unknown with distributions $Z$ ~ $N(0,\sigma_z^2)$ and $X$ ~ $N(0,\sigma_x^2)$. So Y is known but its components are not. I guess that means X and Y are not independent.
Sep 12, 2011 at 7:02 history answered fabee CC BY-SA 3.0