Convergence of Bernoulli distribution using Central Limit Theorem

Question

while I was studying the Central Limit theorem, this problem showed up:

If $0<p = 1-q < 1$ and $x > 0$, then $$ \sum_{k \in K} \binom{n}{k}p^kq^{n-k} \to 2 \int_{0}^{x} \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}u^2} du, $$ where $K := \left\{k: np - x\sqrt{npq} \leq k \leq np+x\sqrt{npq}\right\}$.

The book sugests to apply the theorem to a sequence of random variables with the Bernoulli distribuition. Can somebody help me with this? Doesn't look hard, but I cant understand even the first steps.

Thanks in advance!!

Answer 1

Let $(X_i)$ be an i.i.d sequence where $X_n$ takes the value $1$ with probability $p$ and the value $0$ with probability $q$. Let $S_n=X_1+X_2+\cdots+X_n$. Note that $EX_n=p$ and $EX_n^{2}=p$ so the variance of $X_n $ is $p-p^{2}=pq$. By CLT $P(-x \leq \frac {S_n-np} {\sqrt {npq}} \leq x) \to \int_{-x}^{x} \frac 1{\sqrt {2\pi}} e^{-x^{2}/2} dx=2\int_{0}^{x} \frac 1{\sqrt {2\pi}} e^{-x^{2}/2} dx$. Now just re-write the event on the left.