Question: Let $X_1, X_2, ...$ be a sequence of iid $\textrm{Poi}(5)$ variables. Let $Y_1, Y_2, ...$ be a sequence of iid $\textrm{Bin}(10,\dfrac{1}{2})$ variables. Define the sums $S_n = \sum_1^n X_i$ and $T_n = \sum_1^n Y_i$ for $i=1,2,...$.
Calculate asymptotic approximations to these quantities: $$\mathbb{P}(5 n-\sqrt{n}<T_n<5 n + \sqrt n)$$ and $$\mathbb{P}(5 n-\sqrt{n}<S_n<5 n + \sqrt n).$$ Establish which is closer to 1 for large n.
My attempt: I have: $\mathbb{E}(S_n) = 5n$, $\mathbb{E}(T_n)=5n$, $\textrm{Var}(S_n)=5n$, $\textrm{Var}(T_n)=\dfrac{5}{2}n$. $$\mathbb{P}\left(\dfrac{-1}{\sqrt{5}}<\dfrac{S_n - 5 n}{\sqrt{5n}}< \dfrac{1}{\sqrt{5}}\right)\approx 2\Phi\left(\dfrac{1}{\sqrt{5}}\right)-1.$$
Also, $$\mathbb{P}\left(\dfrac{-1}{\sqrt{5/2}}<\dfrac{T_n - 5 n}{\sqrt{5n/2}}< \dfrac{1}{\sqrt{5/2}}\right)\approx 2\Phi\left(\sqrt{\dfrac{2}{5}}\right)-1.$$
Thus, $$\mathbb{P}(5 n-\sqrt{n}<T_n<5 n + \sqrt n)$$ is closer to 1 for large $n$.
Would this be correct? I am worried as I got numerical answers for the last part rather than an expression which tends to 1, which I am expecting due to the Weak Law of Large Numbers? Thank you.