I'm looking at a random variable that takes vectors $\newcommand{\vv}{\mathbf{v}} \vv_1, \dotsc, \vv_n \in \mathbb{R}^d$ and calculates their average, after applying "blankout" noise to them. So we have independent Bernoulli random variables $\xi_1, \dotsc, \xi_n$ with parameters $p(\xi_i = 1) = p_i$. I multiply the $\newcommand{\vv}{\mathbf{v}} \vv_i$ by the $\xi_i$ and take the average of the ones that remain:
$$\newcommand{\vv}{\mathbf{v}} \frac{\xi_1 \vv_1 + \dotsb + \xi_n \vv_n}{\xi_1 + \dotsb + \xi_n}.$$
(I could define it to be zero in case all $\xi_i = 0$.)
What is the variance of such a thing? If you like we can take all $p_i$ equal to simplify the analysis.
I asked about the mean here.