Let $x_1,\ldots,x_n$ be i.i.d. Bernoulli random variables with parameter $1/2$. Let $S=\sum_{i=1}^nx_i$. Using the Central Limit Theorem, show that $ \frac{|2S-n|}{\sqrt n} $ is convergent to a standard normal random variable.
Thank you.
Let $x_1,\ldots,x_n$ be i.i.d. Bernoulli random variables with parameter $1/2$. Let $S=\sum_{i=1}^nx_i$. Using the Central Limit Theorem, show that $ \frac{|2S-n|}{\sqrt n} $ is convergent to a standard normal random variable.
Thank you.
The expectation of $S$ is $\frac{n}{2}$ and its variance is $\frac{n}{4}$ so $\frac{S - \frac{n}{2}}{\sqrt{\frac{n}{4}}} = \frac{2S - n}{\sqrt{n}}$ converges to a standard normal distribution by the Central Limit Theorem.
The absolute value of a normal distribution with mean $0$ is a half-normal distribution so $\frac{|2S - n|}{\sqrt{n}}$ converges to a half-normal distribution with mean $\sqrt{\frac{2}{\pi}}$.