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The following questions are on my homework, I’m looking for someone’s help, not the answers.
Let $A$ and $B$ be non-empty subsets of some set $S$. We define a function $f$ as follows. $$f(A,B) = \begin{cases} |A\cap B|, & \text{if } A\cap B
eq \emptyset \\ 0 & \text{if } A\cap B = \emptyset \end{cases} $$ Prove that $f$ is a nonnegative function and that $\max\left(\frac{1}{|S|} \sum_{A \subseteq S} f(A,S \backslash A) \right) = h(S)$.
Let $S$ be a finite nonempty set. Let $\{|S| = k\}$ denote the event that $|S| = k$. Show that the expected value of $f(A,S \backslash A)$ is $$\frac{k!}{(k-1)!} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{k}\right)$$ when $A \subseteq S$ and 0 when $A = \emptyset$.
Let $X_1, X_2,…, X_n$ be $n$ independent random variables each taking values in the finite set $S$, such that $\sum_{i = 1}^n \mathbb{P}(X_i = s) = k$ for
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