$\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}
