Recall that a fair coin is one where we have equal probability of flipping a heads or a tails; that is, $P(H) = P(T) = 1/2$.
Let $D_n$ be a random variable counting the difference in the number of heads and number of tails after n flips of a fair coin. That is, $D_n = H_n − T_n$ where $H_n$ is the random variable counting the number of heads after n flips of a fair coin, and $T_n$ is defined similarly for tails.
What is the expected value of $D_n$?
What is the variance of $D_n$? The following identity may be useful: $V(aX+b) = a^{2}V(X).$
I have found the expected value of $D_n$ to be 0 but am having some trouble calculating the variance of $D_n$ and was wondering if I could get some help with that.
If there are $n$ flips, then $H_n + T_n = n$ for any possible outcomes of $H_n$ and $T_n$. For example, if $n = 10$ flips, and you observe $H_{10} = 7$, then you already know $T_{10} = 3$ because you can only get heads or tails on any given coin toss.
So if $H_n$ and $T_n$ add up to $n$ always, then you can express $D_n$ in terms of only one of these two random variables; e.g., $$D_n = H_n - (n - H_n) = 2H_n - n.$$ So to use the example, we have $D_n = 7 - 3 = 4$, or we can also calculate it as $D_n = 2(7) - 10 = 14 - 10 = 4.$
Now how do you compute the expectation and variance of $2H_n - n$?