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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.

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You don't want the absolute values there. – Robert Israel Jul 11 '12 at 2:24
I am interesting in the result with absolute values. Is it possible to show? – AlexK Jul 11 '12 at 2:34
With absolute values you get convergence to a half-normal distribution – Henry Jul 11 '12 at 2:41
up vote 1 down vote accepted

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 $\sqrt{\frac{2}{\pi}}$.

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