第2講確率分布

$t$ 分布

X_{1}, \dots, X_{n} \sim N (0, 1), i . i . d .

$\begin{equation} X_1, \ldots, X_n \sim N(0,1),~i.i.d. \end{equation}$

Y = X_{1}^{2} + \dots + X_{n}^{2} \sim χ^{2} (n)

$\begin{equation} Y = X_{1}^{2} + \cdots + X_{n}^{2}~\sim~\chi^{2}(n) \end{equation}$

Z \sim N (0, 1)

$\begin{equation} Z \sim N(0,1) \end{equation}$

のとき，

T = \frac{Z}{\sqrt{\frac{Y}{n}}} \sim t (n)

$\begin{equation} T = \frac{Z}{\ds\sqrt{\frac{Y}{n}}} \sim t(n) \end{equation}$

f (x) = \frac{1}{\sqrt{n} B (\frac{n}{2}, \frac{1}{2})} {(1 + \frac{x^{2}}{n})}^{- \frac{n + 1}{2}}

$\begin{equation} f(x) = \dfrac{1}{\sqrt{n}\,B\left(\dfrac{n}{2},\dfrac{1}{2}\right)}\left(1+\frac{x^2}{n}\right)^{-\frac{n+1}{2}} \end{equation}$

E [T] = 0

$\begin{equation} E[T] = 0 \end{equation}$

V [T] = E [T^{2}] = {\begin{cases} \frac{n}{n - 2} & (n > 2) \\ \infty & (n = 1, 2) \end{cases}

$\begin{equation} V[T] = E[T^{2}] = \left\{ \begin{array}{l} \dfrac{n}{n-2} & (n > 2) \\ \infty & (n=1,2) \end{array} \right. \end{equation}$

Z = \frac{\bar{X} - μ}{σ / \sqrt{n}} \sim N (0, 1)

$\begin{equation} Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1) \end{equation}$

Y = \frac{(n - 1) U^{2}}{σ^{2}} \sim χ^{2} (n - 1)

$\begin{equation} Y = \frac{(n-1)U^2}{\sigma^2} \sim \chi^2(n-1) \end{equation}$

U^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - \bar{X})^{2}

$\begin{equation} U^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2 \end{equation}$

\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}

$\begin{equation} \bar{X} = \frac{1}{n}\sum_{i=1}^{n}X_i \end{equation}$

のとき，

T = \frac{Z}{\sqrt{\frac{Y}{n - 1}}} = \frac{\bar{X} - μ}{\sqrt{\frac{U^{2}}{n}}} \sim t (n - 1)

$\begin{equation} T = \frac{Z}{\ds\sqrt{\frac{Y}{n-1}}} = \frac{\bar{X} - \mu}{\ds\sqrt{\frac{U^2}{n}}} \sim t(n-1) \end{equation}$