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t-distribution

2024/01/05に公開

Given

\begin{align*} Y &\sim \mathcal{N}(y|0, 1) \\ Z &\sim \chi_n^2(z) \\ \end{align*}

let

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

then X follows t-distribution t_n(x) with n digree of freedom.

\begin{align*} X \sim t_n(x) = \frac{1}{\sqrt{n} \Beta(\frac{n}{2}, \frac{1}{2})} \left( \frac{x^2}{n} + 1 \right)^{- \frac{n+1}{2}} \end{align*}

Let U = Z,

\begin{align*} 1 = \int f_{XU}(x, u) dx du &= \int f_{YZ}(y, z) dy dz \\ &= \int f_Y(y) f_Z(z) dy dz \\ &= \int_{-\infty}^\infty \int_0^\infty \frac{1}{\sqrt{2 \pi}} e^{-\frac{y^2}{2}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} z^{\frac{n}{2}-1} e^{-\frac{z}{2}} dy dz \end{align*}

y = x \sqrt{z/n} = x \sqrt{u/n} , z = u,
x: -\infty \rightarrow \infty , u: 0 \rightarrow 1.

\begin{align*} J &= \begin{vmatrix} dy/dx & dy/du \\ dz/dx & dz/du \end{vmatrix} \\ &= \begin{vmatrix} \sqrt{u/n} & \frac{x}{n} \frac{1}{2} (u/n)^{-\frac{1}{2}} \\ 0 & 1 \end{vmatrix} \\ &= \sqrt{u/n} \end{align*}

therefore

\begin{align*} &= \int_{-\infty}^\infty \int_0^\infty \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2 \frac{u}{n}}{2}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} u^{\frac{n}{2}-1} e^{-\frac{u}{2}} \sqrt{u/n} dx du \\ &= \int_{-\infty}^\infty \frac{1}{\sqrt{2 \pi n}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} \int_0^\infty u^{\frac{n -1 }{2}} e^{- \left( \frac{x^2 u}{2n} +\frac{u}{2} \right)} du dx\\ \end{align*}

let r = \frac{x^2 u}{2n} + \frac{u}{2}, u = \frac{2 n}{x^2 + n} r,
du = \frac{2 n}{x^2 + n} dr , r: 0 \rightarrow \infty.

\begin{align*} &= \int_{-\infty}^\infty \frac{1}{\sqrt{2 \pi n}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} \int_0^\infty (\frac{2 n}{x^2 + n} r)^{\frac{n -1 }{2}} e^{- r} \frac{2 n}{x^2 + n} dr dx\\ &= \int_{-\infty}^\infty \frac{1}{\sqrt{2 \pi n}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} (\frac{2 n}{x^2 + n})^{\frac{n + 1 }{2}} \int_0^\infty r^{\frac{n + 1}{2} - 1} e^{- r} dr dx\\ &= \int_{-\infty}^\infty \frac{1}{\sqrt{2 \pi n}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} 2^{\frac{n + 1 }{2}} (\frac{1}{x^2/n + 1})^{\frac{n + 1 }{2}} \Gamma(\frac{n+1}{2}) dx \\ &= \int_{-\infty}^\infty \frac{1}{\sqrt{2 \pi n}} \frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} 2^{\frac{n + 1 }{2}} (\frac{1}{x^2/n + 1})^{\frac{n + 1 }{2}} \frac{\Gamma(\frac{n}{2}) \Gamma(\frac{1}{2})}{\Beta(\frac{n}{2}, \frac{1}{2})} dx \\ &= \int_{-\infty}^\infty \frac{1}{\sqrt{n} \Beta(\frac{n}{2}, \frac{1}{2})} \left( \frac{1}{x^2/n + 1} \right)^{\frac{n + 1 }{2}} dx \\ &= \int_{-\infty}^\infty t_n(x) dx \\ \end{align*}

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