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F-distribution

2024/01/06に公開

Given two independent random variables Y and Z.

\begin{align*} Y &\sim \chi_m^2(Y=y) \\ Z &\sim \chi_n^2(Z=z) \\ \end{align*}

let

\begin{align*} X = \frac{Y/m}{Z/n} \end{align*}

then, X follows a f-distribution F_{m,n}(X=x) with (m, n) degrees of freedom.

\begin{align*} X &\sim F_{m,n}(X=x) \\ &= \frac{m^{m/2} n^{n/2}}{\Beta(m/2, n/2)} \frac{x^{m/2 - 1}}{(mx + n)^{(m+n)/2}} \end{align*}

let U = Z,

\begin{align*} 1 &= \int f_X(x) dx \\ &= \int \int f_{XU}(x, u) du dx \\ &= \int \int f_{YZ}(y, z) dy dz \\ &= \int \int f_Y(y) f_Z(z) dy dz \\ \end{align*}

y = \frac{m z x}{n} = \frac{m u x}{n} , z = u , x: 0 \rightarrow \infty , u: 0 \rightarrow \infty.

The Jaccobian is

\begin{align*} |J| &= \begin{vmatrix} dy / dx & dy / du \\ dz / dx & dz / du \\ \end{vmatrix} \\ &= \begin{vmatrix} \frac{m u}{n} & \frac{m x}{n} \\ 0 & 1 \\ \end{vmatrix} \\ &= \frac{m u}{n} \end{align*}

therefore,

\begin{align*} &= \int_0^\infty \int_0^\infty f_Y(\frac{m u x}{n}) f_Z(u) \frac{m u}{n} dx du \\ &= \int_0^\infty \int_0^\infty \frac{1}{2^{\frac{m}{2}} \Gamma(\frac{m}{2})} (\frac{m u x}{n})^{\frac{m}{2}-1} e^{-\frac{\frac{m u x}{n}}{2}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} u^{\frac{n}{2}-1} e^{-\frac{u}{2}} \frac{m u}{n} dx du \\ &= \int_0^\infty \frac{1}{ 2^{\frac{m+n}{2}} \Gamma(\frac{m}{2}) \Gamma(\frac{n}{2}) } (\frac{m}{n})^{\frac{m}{2}} x^{\frac{m}{2}-1} \int_0^\infty u^{\frac{m+n}{2}-1} e^{-\frac{m u x + n u}{2n}} du dx \\ \end{align*}

let \frac{(mx + n)u}{2n} = t, u = \frac{2 n t}{mx + n}, t: 0 \rightarrow \infty, du = \frac{2n}{m x + n} dt.

\begin{align*} &= \int_0^\infty \frac{1}{ 2^{\frac{m+n}{2}} \Gamma(\frac{m}{2}) \Gamma(\frac{n}{2}) } (\frac{m}{n})^{\frac{m}{2}} x^{\frac{m}{2}-1} \int_0^\infty (\frac{2 n t}{mx + n})^{\frac{m+n}{2}-1} e^{-t} \frac{2n}{m x + n} dt dx \\ &= \int_0^\infty \frac{1}{ \Gamma(\frac{m}{2}) \Gamma(\frac{n}{2}) } (\frac{m}{n})^{\frac{m}{2}} x^{\frac{m}{2}-1} (\frac{n}{mx + n})^{\frac{m+n}{2}} \int_0^\infty t^{\frac{m+n}{2}-1} e^{-t} dt dx \\ &= \int_0^\infty \frac{1}{ \Gamma(\frac{m}{2}) \Gamma(\frac{n}{2}) } (\frac{m}{n})^{\frac{m}{2}} x^{\frac{m}{2}-1} (\frac{n}{mx + n})^{\frac{m+n}{2}} \Gamma(\frac{m}{2} + \frac{n}{2}) dx \\ &= \int_0^\infty \frac{1}{\Beta(\frac{m}{2}, \frac{n}{2})} (\frac{m}{n})^{\frac{m}{2}} x^{\frac{m}{2}-1} (\frac{n}{mx + n})^{\frac{m+n}{2}} dx \\ &= \int_0^\infty \frac{1}{\Beta(\frac{m}{2}, \frac{n}{2})} m^{\frac{m}{2}} n^{\frac{n}{2}} x^{\frac{m}{2}-1} \frac{1}{(mx + n)^{\frac{m+n}{2}}} dx \\ &= \int_0^\infty F_{m,n}(x) dx \\ \end{align*}

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