Let $X = (x_1,\ldots,x_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod_{i=1}^n x_i$

Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that *on average* considers only a subsample of $X$?

Weak conditions may be imposed on $F$, for instance we may assume it has finite mean and variance, we may also assume that $\forall i, x_i >0$, but otherwise nothing is known about $F$.