using Central Limit Theorem to approximate a probability with large 'n'

Question

For $i>1$ , let $X_i$ ~ $G_{1/2}$ be distributed Geometrically with parameter $1/2$.

$$ Y_n= \frac{1}{\sqrt n} \sum_{r=1}^n X_r-2 $$

Approximate $P(-1\le Y_n\le 2)$ with large n.($Y_n$ is not properly normalized)

My attempt-
normalised $Y_n = \frac{1}{\sqrt n} \sum_{r=1}^n \frac{(X_r-2)-E[X_r-2]}{V[X-2]*n}$

then approximating $P(-1\le Y_n\le 2)$ I got the answer as 0.4332 which is incorrect. Can anybody give me the correct solution of this question.

Answer 1

Like $X_r-2$, $Y_n$ has mean $0$ and variance $2$. The answer is therefore $\Phi(\sqrt{2})-\Phi(\frac{-1}{\sqrt{2}})\approx 0.6816$.