I'm betting on tosses of a fair coin. If the coin comes up heads I win 1 dollar (probability $p=1/2$), if the coin comes up tails, I lose 1 dollar (probability $q=1/2$).
I know that the number of wins follows a binomial distribution, and that the variance of the number of wins after $n$ tosses is $npq = n/4$.
However, I am interested in the variance of my accumulated winnings, not the number of wins. I know from both literature (see, e.g., Eq. (6) in Chandrashekhar, 1943)) and numerical experimentation that the answer is $n$, but how can I show this?
(An alternative description of the same problem: Consider a random walker, which starts at 0 and takes a step of length 1 to the right with probability $p=1/2$, or a step of length 1 to the left with probability $q=1/2$. What is the mean square displacement after $n$ steps? In this context, I know an argument that shows by induction that the answer is $n$, but I'm looking for a more "statistical" answer that uses information about the binomial distribution.)