Using the binomial distribution to compute the limit of $\sum\limits_{i=\lfloor an \rfloor}^{\lfloor bn \rfloor} {n \choose i } p^i (1-p)^{n-i}$

Question

Knowing that the sum of $n$ independent Bernoulli random variables with parameter $p$ in $(0,1)$ has a binomial distribution Bin$(n,p)$, how to use the central limit theorem (or any other limit theorem in probability) to determine how the following depends on $a$ and $b$ ($0 \leq a < b \leq 1$) $$\lim_{n\to\infty} \sum_{i=\lfloor an \rfloor}^{\lfloor bn \rfloor} {n \choose i } p^i (1-p)^{n-i}\ ?$$

Looking at the graph on the binomial cumulative distribution function, I see how it behaves but I've tried many different limit theorems and I can't produce any proper proof for this.

Answer 1

Hints:

• If $a\ne p\ne b$, use the law of large numbers for sums of i.i.d. random variables $X_k$ with distribution $P(X_k=1)=p$, $P(X_k=0)=1-p$
• If $a=p$ or $b=p$, use the central limit theorem applied to the same i.i.d. sequence

Final answer:

• The limit is $1$ if $a\lt p\lt b$, $\frac12$ if $a=p\lt b$ or if $a\lt p=b$, and $0$ if $a\lt b\lt p$ or if $p\lt a\lt b$.