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$$\bar{x}_n=\frac{x_1+x_2+\cdots x_n}{n},\hspace{0.5zw} y_n=\sqrt{n}\frac{\bar{x}_n-\mu}{\sigma}$$

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$$\sigma\left(\begin{array}{c} n \\ x \end{array}\right)p^xq^{n-x}\rightarrow\frac{1}{\sqrt{2\pi}}e^{-u^2/2}$$

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$$n!\sim n^n\sqrt{2\pi n}e^{-n}\hspace{0.5zw}(n\rightarrow\infty)$$ (1)

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$$n!= O(n^n\sqrt{2\pi n}e^{-n}),\hspace{1zw} n!= n^n\sqrt{2\pi n}e^{-n}(1+o(1))$$

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$$\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri... ... npq + \log n! - \log x! - \log (n-x)! + x\log p + (n-x)\log q \end{eqnarray*}$$

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$$\begin{eqnarray*}x & = & np+u\sqrt{npq} = n\left(p+u\sqrt{\frac{pq}{n}}\right) ... ...pq} = n\left(q-u\sqrt{\frac{pq}{n}}\right) \rightarrow\infty \end{eqnarray*}$$

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$$\begin{eqnarray*}\log m! & = & \log m^m\sqrt{2\pi m}e^{-m}(1+o(1)) \\ & = & m\... ...frac{1}{2}\log 2\pi + \left(m+\frac{1}{2}\right)\log m -m +o(1) \end{eqnarray*}$$

$B$G$"$k$N$G!"(B

$$\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri... ...frac{n}{n-x} +x\log p\frac{n}{x}+(n-x)\log q\frac{n}{n-x}+o(1) \end{eqnarray*}$$

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$$\frac{x}{n} = p + u\sqrt{\frac{pq}{n}},\hspace{1zw} \frac{n-x}{n} = q - u\sqrt{\frac{pq}{n}}$$

$B$G$"$k$N$G!"(B

$$\log p\frac{n}{x} \rightarrow \log 1 = 0,\hspace{1zw} \log q\frac{n}{n-x} \rightarrow \log 1 = 0$$

$B$H$J$k$N$G(B

$$\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri... ...p}}\right) -(n-x)\log \left(1-u\sqrt{\frac{p}{nq}}\right)+o(1) \end{eqnarray*}$$

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$$\log(1+x)=x-\frac{x^2}{2}+O(x^3)\hspace{1zw}(x\rightarrow 0)$$

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$$\begin{eqnarray*}\lefteqn{x\log\left(1+u\sqrt{\frac{q}{np}}\right) = (np+u\sqrt... ...-u\sqrt{npq}+\frac{u^2}{2}p+O\left(\frac{1}{\sqrt{n}}\right)\\ \end{eqnarray*}$$

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$$\begin{eqnarray*}\lefteqn{\log\sigma\left(\begin{array}{c} n \\ x \end{array}\ri... ...\log 2\pi -\frac{u^2}{2} = \log\frac{1}{\sqrt{2\pi}}e^{-u^2/2} \end{eqnarray*}$$