I am trying to understand the mathematical aspects of a Galton board with (as) "elementary" (as possible) tools.
I had the idea that the gaussian shape which appears with the board can be explained by the convergence in distribution of a normalized sum of binomial variables to a $\mathcal{N}(0,1)$ variable (CLT). Am I right (especially for the type of convergence) ?
For the mathematical aspect : let $X_1,\dots,X_n$ $n$ random independent variables with the same distribution $\mathcal{B}(n,p)$ where $0<p<1$. If $x\in\mathbb{R}$, the CLT implies $\displaystyle\mathbb{P}\left(\frac{S_n-np}{\sqrt{np(1-p)}}\leq x\right)\underset{{n}\longrightarrow{+\infty}}\longrightarrow\frac 1{\sqrt{2\pi}}\int_{-\infty}^x\mathrm{e}^{-\frac{t^2}2}\mathrm{d}t$. I would like to prove this limit by "elementary" means (Stirling approximation, integral representations and dominated convergence for example).
I suppose that $p=\frac 12$ for simplicity. We have to prove $\displaystyle\sum_{j=0}^{\lfloor np+x\sqrt{np(1-p)}\rfloor}\mathbb{P}\left(S_n=j\right)=\frac 1{2^n}\sum_{j=0}^{\lfloor\frac{n+x\sqrt n}2\rfloor}\binom{n}{j}\underset{{n}\longrightarrow{+\infty}}\longrightarrow\frac 1{\sqrt{2\pi}}\int_{-\infty}^x\mathrm{e}^{-\frac{t^2}2}\mathrm{d}t$.
It can be shown that $\displaystyle\sum_{j=k}^n\binom njp^j(1-p)^{n-j}=k\binom nk\int_0^pt^{k-1}(1-t)^{n-k}\mathrm{d}t$.
My second question is : how can I find the limit of $\displaystyle k\binom nk\int_0^{\frac 12}t^{k-1}(1-t)^{n-k}\mathrm{d}t$ as $n\longrightarrow +\infty$, where $k=\lfloor\frac{n+x\sqrt n}2\rfloor+1$ ? Pearhaps the Laplace method could conclude, but I don't find a good change of variable and would like to apply dominated convergence theorem... Thank you.