Since the [[Binomial Distribution]] can be viewed as the sample sum of repeated [[Bernoulli Distribution|Bernoulli]] [[Random Variable|random variables]], it satisfied the assumptions of the [[Central Limit Theorem]]. Thus, a r.v. $X\sim \mathrm{Binom}(n,p)$ can be approximated by a [[Gaussian Distribution]], after being standardized. **Binomial Distribution:** $ p_X(X=k) =\sum_{k=0}^{n} \binom{n}{k} * (1-p)^{n-k}* p^n $ **Approximation as Standard Gaussian:** From the properties of the Binomial, we know the [[Expectation]] and [[Variance]], to replace $\mu$ and $\sigma$. $ Z_n=\frac{X -\mu}{\sigma} \xrightarrow[n \to \infty]{(d)} \mathcal N(0,1) \quad \text{where} \begin{cases} \mu=np\\ \sigma^2= np*(1-p) \end{cases} $ In terms of binomial parameters: $ Z_n=\frac{X-np}{\sqrt{np*(1-p)}} \xrightarrow[n \to \infty]{(d)} \mathcal N(0,1) $ ## De Moivre-Laplace Correction While both the Binomial PMF and the Gaussian approximation yield similar results for large $n$, we need to correct for the fact that the latter is a continuous distribution. Therefore we sum integrate over the interval $[k-0.5, k+0.5]$ to get the probability mass $p_X(X=k)$ at a specific $k$. **Example:** Assess the probability of $k=19$, when $X \sim \mathrm{Binom}(n=36; p=0.5)$ via Gaussian approximation. ![[de-moivre-laplace.png|center|400]] $ \begin{aligned} \mathbf P(k=19) &= \mathbf P(18.5 \le S_n \le 19.5) \\[8pt] &=\mathbf P\Big(\frac{18.5-18}{3} \le Z_n \le \frac{19.5-18}{3} \Big) \\[8pt] &=\mathbf P(0.17 \le Z_n \le 0.5) \\[8pt] &=\Phi(0.5) - \Phi(0.17) \\[8pt] &=0.124 \end{aligned} $