Calculating entropy for a given problem

Question

Let $X_1,X_2,\dots$ be an i.i.d. sequence of binary random variables, each equally likely to be $0$ or $1$.

We define:

$Y_n$ := $|\{j : 1 \le j \le n, X_j = 1\}|$, $n = 1,2,\dots$, where $|.|$ denotes the number of the elements of the set.

How can we calculate $H(Y)$?

Answer 1

Note that $$P(Y_n=i)=\binom{n}{i}2^{-n},$$ and hence you can compute the entropy.