I am looking for a reference (e.g. in some textbook) of the following:
Let $F$ be a finite set, and $X$ a random subset of $F$ (i.e., a random variable with values in the power set of $F$) such that $\mathbb{E}(|X|)=\alpha |F|$, for some $0<\alpha<1$. Here $|\cdot|$ denotes the cardinality of the set. Then $$H(X)\le |F|(-\alpha \log \alpha - (1-\alpha)\log(1-\alpha)).$$
It can be proven quickly using properties of the entropy function, but I would really like to find a reference that I could cite.
Thank you!