第1講確率と確率変数

中心極限定理

$X_{1},~X_{2},~\ldots,X_{n}$ は独立で同分布とし， $E[X_i] = \mu,~V[X_i] = \sigma$ とする．また，

\bar{X} = \frac{X_{1} + X_{2} + \dots + X_{n}}{n} (E [\bar{X}] = μ, V [\bar{X}] = \frac{σ^{2}}{n})

$\begin{equation} \bar{X} = \frac{X_{1} + X_{2} + \cdots + X_{n}}{n}~~~~(E[\bar{X}]=\mu,~V[\bar{X}]=\frac{\sigma^2}{n}) \end{equation}$

とし，

\bar{X}

$\bar{X}$ を標準化したものを

{\bar{X}}^{*}

$\bar{X}^{*}$ と書くことにすると，

{\bar{X}}^{*} = \frac{X_{1} + X_{2} + \dots + X_{n} - n μ}{\sqrt{n} σ} \sim N (0, 1) (n \to \infty)

$\begin{equation} \bar{X}^{*} = \frac{X_{1} + X_{2} + \cdots + X_{n} - n\mu}{\sqrt{n}\sigma} \sim N(0,1)~~~~(n \to \infty) \end{equation}$

補足1: 中心極限定理の証明

\begin{aligned} M_{{\bar{X}}^{*}} (α) \\ = M_{\frac{X_{1} - μ}{\sqrt{n} σ} + \frac{X_{2} - μ}{\sqrt{n} σ} + \dots + \frac{X_{n} - μ}{\sqrt{n} σ}} (α) \\ = E [e^{\frac{α}{\sqrt{n}} (\frac{X_{1} - μ}{σ} + \frac{X_{2} - μ}{σ} + \dots + \frac{X_{n} - μ}{σ})}] \\ = E [e^{α \frac{X_{1} - μ}{\sqrt{n} σ}} e^{α \frac{X_{2} - μ}{\sqrt{n} σ}} \dots e^{α \frac{X_{n} - μ}{\sqrt{n} σ}}] \\ = {E [e^{α \frac{X_{1} - μ}{\sqrt{n} σ}}]}^{n} \\ = {1 + \frac{1}{1!} \cdot E [{(\frac{X_{1} - μ}{\sqrt{n} σ})}^{1}] \cdot α^{1} + \frac{1}{2!} \cdot E [{(\frac{X_{1} - μ}{\sqrt{n} σ})}^{2}] \cdot α^{2} + \dots}^{n} \\ = {1 + \frac{1}{1!} \cdot \frac{1}{\sqrt{n}} \cdot E [{(\frac{X_{1} - μ}{σ})}^{1}] \cdot α^{1} + \frac{1}{2!} \cdot \frac{1}{n} \cdot E [{(\frac{X_{1} - μ}{σ})}^{2}] \cdot α^{2} + \dots}^{n} \\ = {1 + \frac{1}{1!} \cdot \frac{1}{\sqrt{n}} \cdot 0 \cdot α^{1} + \frac{1}{2!} \cdot \frac{1}{n} \cdot 1 \cdot α^{2} + \dots}^{n} \\ = {1 + \frac{α^{2}}{2 n} + \dots}^{n} \end{aligned}

$\begin{align} & M_{\bar{X}^{*}}(\alpha) \\ &\quad = M_{\frac{X_1-\mu}{\sqrt{n}\sigma} + \frac{X_2-\mu}{\sqrt{n}\sigma} + \cdots + \frac{X_n-\mu}{\sqrt{n}\sigma}}(\alpha) \\ &\quad = E\left[e^{\frac{\alpha}{\sqrt{n}}(\frac{X_1-\mu}{\sigma} + \frac{X_2-\mu}{\sigma} + \cdots + \frac{X_n-\mu}{\sigma})}\right] \\ &\quad = E\left[e^{\alpha\frac{X_1-\mu}{\sqrt{n}\sigma}} e^{\alpha\frac{X_2-\mu}{\sqrt{n}\sigma}} \cdots e^{\alpha\frac{X_n-\mu}{\sqrt{n}\sigma}}\right] \\ &\quad = \left\{E\left[e^{\alpha\frac{X_1-\mu}{\sqrt{n}\sigma}}\right]\right\}^n \\ &\quad = \left\{1 + \frac{1}{1!}\cdot E\left[\left(\frac{X_1-\mu}{\sqrt{n}\sigma}\right)^1\right]\cdot \alpha^1 + \frac{1}{2!}\cdot E\left[\left(\frac{X_1-\mu}{\sqrt{n}\sigma}\right)^2\right]\cdot \alpha^2 + \cdots \right\}^n \\ &\quad = \left\{1 + \frac{1}{1!}\cdot\frac{1}{\sqrt{n}}\cdot E\left[\left(\frac{X_1-\mu}{\sigma}\right)^1\right]\cdot \alpha^1 + \frac{1}{2!}\cdot\frac{1}{n}\cdot E\left[\left(\frac{X_1-\mu}{\sigma}\right)^2\right]\cdot \alpha^2 + \cdots \right\}^n \\ &\quad = \left\{1 + \frac{1}{1!}\cdot\frac{1}{\sqrt{n}}\cdot 0 \cdot \alpha^1 + \frac{1}{2!}\cdot\frac{1}{n}\cdot 1 \cdot \alpha^2 + \cdots \right\}^n \\ &\quad = \left\{1 + \frac{\alpha^2}{2n} + \cdots \right\}^n \end{align}$

よって，

\begin{aligned} lim_{n \to \infty} M_{{\bar{X}}^{*}} (α) \\ = lim_{n \to \infty} {1 + \frac{α^{2}}{2 n} + \dots}^{n} \\ = e^{\frac{α^{2}}{2}} = M_{N (0, 1)} (α) \end{aligned}

$\begin{align} & \lim_{n \to \infty}M_{\bar{X}^{*}}(\alpha) \\ &\quad = \lim_{n \to \infty}\left\{1 + \frac{\alpha^2}{2n} + \cdots \right\}^n \\ &\quad = e^{\frac{\alpha^2}{2}} = M_{N(0,1)}(\alpha) \end{align}$

つまり，

\begin{aligned} {\bar{X}}^{*} \to_{n \to \infty}^{d} N (0, 1) (分布収束) \end{aligned}

$\begin{align} \bar{X}^{*} \xrightarrow[n \to \infty]{d} N(0,1)~~~~(\text{分布収束}) \end{align}$

第1講 確率と確率変数

中心極限定理

補足1: 中心極限定理の証明

第1講確率と確率変数