$\S$7.4.1 Normal distribution,chi-squared distribution, Student $t$-distribution,$F$-distribution

Definition 7.6 [$F$distribution ]. Let $t \ge 0$, and $n_1$ and $n_2$ be natural numbers. The probability density function $p_{(n_1,n_2)}^F(t)$ of $F$-distribution with the degree of freedom$(n_1,n_2)$ is defined by

\begin{align} p_{(n_1,n_2)}^F(t) = \frac{1}{B(n_1/2, n_2/2)} \Big(\frac{n_1}{n_2} \Big)^{n_1/2} \frac{t^{(n_1-2)/2}}{(1+n_1t/n_2)^{(n_1+n_2)/2}} \qquad (t \ge 0) \tag{7.71} \end{align}

where, $B(\cdot, \cdot)$ is the Beta function,that is, for $x, y > 0$,

\begin{align} B(x, y ) = \int_0^1 t^{x-1} (1-t)^{y-1} dt \end{align}

Note that

\begin{align} & \mbox{ $F$-distribution with degree of freedom$(1,n-1)$ } \\ = & \mbox{ Student $t$-distribution with the degree of freedom$(n-1)$} \end{align}

Define two maps $\overline{\mu}: {\mathbb R}^n \to {\mathbb R}$ and $\overline{SS}: {\mathbb R}^n \to {\mathbb R}$ as follows.

\begin{align} & \overline{\mu} (x)=\overline{\mu}(x_1, x_2, \cdots, x_n ) = \frac{\sum_{k=1}^n x_k }{n} \\ & \overline{SS} (x)=\overline{SS}(x_1, x_2, \cdots, x_n ) = {\sum_{k=1}^n (x_k - \overline{\mu} (x))^2 } \\ & \qquad( \forall x = (x_1, x_2, \cdots, x_n ) \in {\mathbb R}^n ) \end{align}

Formula 7.7 [Gauss integral(normal distribution and chi-squared distribution)]. This was already mentioned in (6.6) and (6.7).

Formula 7.8 [Gauss integral($F$-distribution )]. For $c \ge 0$,

\begin{align} \;\;\;\; \mbox{(A):$\;\;\;$} & \frac{1}{({{\sqrt{2 \pi }{}}})^n} \underset{ c \le \frac{ n(\overline{\mu}(x))^2 }{ {\overline{SS}(x)}/({n-1}) } } {\int \cdots \int} \exp[{}- \frac{\sum_{k=1}^n ({}{x_k} {} )^2 } {2 } {}] d {}{x_1} d {}{x_2}\cdots dx_n = \int^{\infty}_{ c } p_{(1,{{n}}-1) }^F(t) dt \tag{7.72} \end{align}

$\quad$ (B): For $n=\sum_{i=1}^a n_i$,

\begin{align} & \frac{1}{({{\sqrt{2 \pi }{}}})^{{{n}}}} \underset{ \frac{ (\sum_{i=1}^a n_i( x_{i \bullet} - x_{\bullet \bullet} )^2 /(a-1)}{ (\sum_{i=1}^a \sum_{k=1}^{n_i} (x_{ik} - x_{i \bullet})^2)/({{n}}-a) } > c } {\int \cdots \int} \exp[{}- \frac{ \sum_{i=1}^a \sum_{k=1}^{n_i} ({}{x_{ik}} )^2 } {2 } {}] \times_{i=1}^a \times_{k=1}^{n_i} d {}{x_{ik}} \nonumber \\ = & \int^{\infty}_{c} p_{(a-1,{{n}}-a) }^F(t) dt \tag{7.73} \end{align} \begin{align} \mbox{(C)} \;\; & \frac{1} {({ {\sqrt{2 \pi }} })^{abn}} \underset{ \frac{ \frac{ \sum_{i=1}^a \sum_{j=1}^b( x_{ij \bullet} - x_{\bullet \bullet \bullet} )^2}{(a-1)} }{ \frac{ \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^n (x_{ijk} - x_{ij \bullet})^2 }{ab(n-1)} } > c } {\int \cdots \int} \exp[- \frac{ \sum_{i=1}^a \sum_{j=1}^b \sum_{k=1}^n (x_{ijk} )^2 }{2 } ] \times_{k=1}^n \times_{j=1}^b \times_{i=1}^a d{x_{ijk} } \nonumber \\ = & \int^{\infty}_c p_{(a-1,ab(n-1)) }^F(t) dt \tag{7.74} \end{align}

Or, equivalently

\begin{align} \mbox{(D):$\;\;\;$} & \frac{1} {({ {\sqrt{2 \pi }} })^{abn}} \underset{ \frac{ \frac{ \sum_{i=1}^a\sum_{j=1}^b( x_{i j \bullet } - x_{i \bullet \bullet } - x_{\bullet j \bullet } + x_{\bullet \bullet \bullet } )^2}{(a-1)(b-1)} }{ \frac{\sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^n (x_{ijk} - x_{ij \bullet})^2}{ ab(n-1) } } > c } {\int \cdots \int} \exp[- \frac{ \sum_{i=1}^a \sum_{j=1}^b \sum_{k=1}^n (x_{ijk} )^2 }{2} ] \times_{k=1}^n \times_{j=1}^b \times_{i=1}^a d{x_{ijk} } \nonumber \\ = & \int^{\infty}_{c} p_{((a-1)(b-1),ab(n-1)) }^F (t) dt \tag{7.75} \end{align}