Let $S$ = $\{v\_1, v\_2, ..., v\_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v\_i$ = $\langle x^i\_1x^i\_2... x^i\_d\rangle$ for all $i \in [1,n]$, then what's the expected sum of all $x^i\_1$ (the first "coordinate" of a vertex $v\_i$) for all $i \in [1,n]$? Is it $\frac{n}{2}$?



Expected sum of chosen coordinates in a random subset of a Hamming hypercube
Yes. There is an involution $\phi(x\_1,\cdots,x\_d)=(1-x\_1,\cdots,1-x\_d)$ which can extend to subsets by defining $\phi(S)=\{\phi(v\_i)\vert v\_i\in S\}$. If $X$ is the random variable which takes the value of the sum of all first coordinates then we have
$$\mathbb E[X]=\frac{1}{2}\mathbb E[X+\phi(X)]=\frac{n}{2}$$
as expected.


