第一種オイラー積分 / ベータ凾数

第一種オイラー積分 / ベータ凾数

$p,q\ge 0$の整数の元で以下の式を第一種オイラー積分という． $$\begin{array}{r}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x\end{array}$$ 部分積分を一回適用すると以下のように$p+1$,$q-1$の第一種オイラー積分がでてくる． $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x& =& {\int }_{\alpha }^{\beta }\left[\frac{\mathrm{d}}{\mathrm{d}x}\{\frac{1}{p+1}(x-\alpha {)}^{(p+1)}\}\right](\beta -x{)}^q\mathrm{d}x\cdots \frac{\mathrm{d}}{\mathrm{d}x}\{\frac{1}{p+1}(x-\alpha {)}^{(p+1)}\}=(x-\alpha {)}^p\\ & =& {[\frac{1}{p+1}(x-\alpha {)}^{(p+1)}(\beta -x{)}^q]}_{\alpha }^{\beta }-{\int }_{\alpha }^{\beta }\frac{1}{p+1}(x-\alpha {)}^{(p+1)}\{q(\beta -x{)}^{(q-1)}(-1)\}\mathrm{d}x\cdots \int f^{\prime }g\mathrm{d}x=fg-\int f{g}^{\prime }\mathrm{d}x\\ & =& 0-{\int }_{\alpha }^{\beta }\frac{-q}{p+1}(x-\alpha {)}^{(p+1)}(\beta -x{)}^{(q-1)}\mathrm{d}x\cdots (\alpha -\alpha )=0,(\beta -\beta )=0\\ & =& \frac{q}{p+1}{\int }_{\alpha }^{\beta }(x-\alpha {)}^{(p+1)}(\beta -x{)}^{(q-1)}\mathrm{d}x\end{array}$$ 第一種オイラー積分の乗数$p,q$に着目し元の式を$I_{p,q}$,部分積分によって出てきた積分を$I_{p+1,q-1}$と表すと以下の様になる． $$\begin{array}{rcl}{I}_{p,q}& =& \frac{q}{p+1}{I}_{p+1,q-1}\cdots I_{a,b}={\int }_{\alpha }^{\beta }(x-\alpha {)}^a(\beta -x{)}^b\mathrm{d}x\\ & =& \frac{q}{p+1}\frac{q-1}{p+2}{I}_{p+2,q-2}\\ & =& \frac{q}{p+1}\frac{q-1}{p+2}\cdots \frac{2}{p+(q-1)}\frac{1}{p+q}{I}_{p+q,0}\\ I_{p+q,0}& =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^{(p+q)}(\beta -x{)}^0\mathrm{d}x\\ & =& {\int }_{\alpha }^{\beta }(x-\alpha {)}^{(p+q)}\mathrm{d}x\\ & =& {[\frac{1}{p+q+1}(x-\alpha {)}^{\{(p+q)+1\}}]}_{\alpha }^{\beta }\\ & =& \frac{1}{p+q+1}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{(\beta -\alpha {)}^{(p+q+1)}}{p+q+1}\\ I_{p,q}& =& \frac{q}{p+1}\frac{q-1}{p+2}\cdots \frac{2}{p+q-1}\frac{1}{p+q}\frac{(\beta -\alpha {)}^{(p+q+1)}}{p+q+1}\\ & =& \frac{q(q-1)\cdots 21}{(p+1)(p+2)\cdots (p+q-1)(p+q)(p+q+1)}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{q!}{\frac{(p+q+1)!}{p!}}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\end{array}$$ 特に$\alpha =0,\beta =1$の時 $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x& =& {\int }_0^1(x-0{)}^p(1-x{)}^q\mathrm{d}x\\ & =& {\int }_0^1{x}^p(1-x{)}^q\mathrm{d}x\\ & =& \frac{p!q!}{(p+q+1)!}(1-0{)}^{(p+q+1)}\cdots {\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}(\beta -\alpha {)}^{(p+q+1)}\\ & =& \frac{p!q!}{(p+q+1)!}\end{array}$$ 特に$\alpha =0,\beta =1,p=m-1,q=n-1$の時 $$\begin{array}{rcl}{\int }_{\alpha }^{\beta }(x-\alpha {)}^p(\beta -x{)}^q\mathrm{d}x& =& {\int }_0^1{x}^{(m-1)}(1-x{)}^{(n-1)}\mathrm{d}x\\ & =& \frac{(m-1)!(n-1)!}{((m-1)+(n-1)+1)!}\cdots {\int }_0^1{x}^p(1-x{)}^q\mathrm{d}x=\frac{p!q!}{(p+q+1)!}\\ & =& \frac{(m-1)!(n-1)!}{(m+n-1)!}\\ & =& \frac{\mathrm{\Gamma }(m)\mathrm{\Gamma }(n)}{\mathrm{\Gamma }(m+n)}\cdots \mathrm{\Gamma }(a)=(a-1)!\\ & =& B(m,n)\cdots \mathrm{ベ}\mathrm{ー}\mathrm{タ}\mathrm{凾}\mathrm{数}\end{array}$$ 逆を言えば上記積分の値を1にしたい時はベータ凾数で割れば(逆数を掛ければ)よい(確率密度凾数との関係性)． $$\begin{array}{rcl}\frac{1}{B(m,n)}{\int }_0^1{x}^{(m-1)}(1-x{)}^{(n-1)}\mathrm{d}x& =& 1\\ f(x;m,n)& =& \frac{x^{(m-1)}(1-x{)}^{(n-1)}}{B(m,n)}\cdots \mathrm{ベ}\mathrm{ー}\mathrm{タ}\mathrm{分}\mathrm{布}\end{array}$$