Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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: enter image description here

share|cite|improve this question
up vote 1 down vote accepted

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 $X_i$.

share|cite|improve this answer
Could you elaborate on how you found that? – HCAI Aug 9 '12 at 14:36
@user1134241: which part? – joriki Aug 9 '12 at 14:51
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
The latter I think is probably it. Thanks – HCAI Aug 9 '12 at 16:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.