using clt to approximate probability

Question

Given $X~Binomial(n,1/4)$ I need to calculate $P(X/\sqrt{n}<0.5)$. But I cannot figure out a way to get rid of $\sqrt{n}$. Should I apply CLT using the mean and variance of $X/\sqrt{n}$ ?

Answer 1

Actually, you should not get rid of $\sqrt n$, as it is precisely what you need to make appear something linked to the central limit theorem. The other ingredient is the following: you can see the random variable $X$ as a sum of $n$ independent Bernoulli random variables $X_i$ (taking the value one with probability $1/4$ and $0$ with probability $1-1/4$.

Therefore, the probability you want to approximate is $$ \mathbb P\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i<0.5\right). $$ You are not yet ready to apply the central limit theorem as the random variables $X_i$ are not centered hence you have to substract the expectations.