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: