We are given a set $S=\{1, 2, \ldots, n\}$ where $n\gg 1$, and for all indices $1\le i \le n$, $i$ is associated with a real value $\alpha_i\!\cdot\! v_i$, where $\alpha_i\in[0,1]$ and $v_i\in(0,1]$.

Let $X$ be a discrete random variable whose sample space consists of $n'<n$ (not necessarily disjoint) proper subsets of $S$. We denote all the possible outcomes as $S_1, S_2, \ldots, S_{n'}$. We also have $\bigcup_{1\le k\le n'} S_k=S$, and the values taken by $X$ are defined as follows for all $1\le k\le n'$:

$$X(S_k)\equiv x_k:=\frac{y_k}{z_k}~,$$

where $y_k$ and $z_k$ are the values taken by the two random variables $Y$ and $Z$ (with the same sample space, outcomes and constraints of $X$), defined as $$Y(S_k)\equiv y_k:=\sum_{j\in S_k}\!\left(\alpha_j\!\cdot\! v_j\right)$$ and $$Z(S_k)\equiv z_k:=\sum_{j\in S_k} v_j~.$$

For $1\le i\le n$, let $p(i):=\sum_{k: i\in S_k} \Pr(S_k)$. Note that we have $\mathbb{E}[Y]=\sum_{1\le i\le n} \left(p(i)\,\alpha_i\,v_i\right)$ and $\mathbb{E}[Z]=\sum_{1\le i\le n} \left(p(i)\,v_i\right)$.

**Question:** Is it possible to express the expectation $\mathbb{E}[X]$ (or a lower bound for it *close* to $\mathbb{E}[X]$ for * all* possible input values of this problem), as a function of $\alpha_1, \alpha_2, \ldots, \alpha_n$,$~~$$v_1, v_2, \ldots, v_n$,$~~$and $~p(1), p(2), \ldots, p(n)$ —

*like above for $\mathbb{E}[Y]$ and $\mathbb{E}[Z]$*?