# How to calculate conditional variance of expectation V(E[Y|X=x_i]) from a data set or function?

I have a function $Y=\displaystyle \sum_{i=1}^n X_i \Omega_i$, which represents the Isotope picked up after surfaces are touched. Here $X_i\sim N (\mu_i,\sigma_i)$. and $\Omega_i$ are constants of surface isotope concentration.

How can I calculate the conditional variance of the expectation of $Y$ given a particular $x_{i^*}$: ie $V_{x_i^*}(E[Y|X_i=x_{i^*}])$?. Can this be done analytically?

The data set produces for example:

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Unless I'm misunderstanding the question, this is straightforward: Expectation is linear, so the expectation of all terms except for the $i^*$-th just becomes a constant that doesn't affect the variance, so this is just $V_{x_i^*}(E[X_i\Omega_i|X_i=x_{i^*}])$, which is $\Omega_i^2$ times the variance of $X_i$.
The analytical finding of the variance if possible.I'm attempting to calculate sensitivity indices $S_i$. If $X_i$ was drawn from data rather than a known continuous distribution, would this alter the outcome? –  HCAI Aug 9 '12 at 15:24
@user1134241: I still don't feel sure I understand your question correctly. There's not much analytical in my answer; I just applied the linearity of expectation and the translational invariance of the variance. I don't see how this could have anything to do with how you obtain the distribution for $X_i$. But I may just not be following what you do. Perhaps you could say more specifically what part of the answer you don't understand. The linearity? The translational invariance? The fact that the variance of $X\Omega$ is $\Omega^2$ times the variance of $X$? –  joriki Aug 9 '12 at 15:29