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

Kavi Rama Murthy · Accepted Answer

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 $P(-x leq frac {S_n-np} {sqrt {npq}} leq x) to int_{-x}^{x} frac 1{sqrt {2pi}} e^{-x^{2}/2} dx=2int_{0}^{x} frac 1{sqrt {2pi}} e^{-x^{2}/2} dx$. Now just re-write the event on the left.

Correct answer by Kavi Rama Murthy on March 4, 2021

Convergence of Bernoulli distribution using Central Limit Theorem

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