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| Boilerpipe Text | Let \(X_1\) and \(X_2\) have independent gamma distributions with parameters \(\alpha, \theta\) and \(\beta\) respectively. Therefore, the joint pdf of \(X_1\) and \(X_2\) is given by \(\begin{align*}
f(x_1, x_2) = \frac{1}{\Gamma(\alpha) \Gamma(\beta)\theta^{\alpha + \beta}} x_1^{\alpha-1}x_2^{\beta-1}\text{ exp }\left( -\frac{x_1 + x_2}{\theta} \right), 0 <x_1 <\infty, 0 <x_2 <\infty.
\end{align*}\) We make the following transformation: \(\begin{align*}
Y_1 = \frac{X_1}{X_1+X_2}, Y_2 = X_1+X_2
\end{align*}\) The inverse transformation is given by \(\begin{align*}
&X_1=Y_1Y_2,
\\& X_2=Y_2-Y_1Y_2
\end{align*}\) The Jacobian is \(\begin{align*}
\left|
\begin{array}{cc}
y_2 & y_1 \\ -y_2 & 1-y_1
\end{array}
\right| = y_2(1-y_1) + y_1y_2 = y_2
\end{align*}\) The joint pdf \(g(y_1, y_2)\) is \(\begin{align*}
g(y_1, y_2) = |y_2| \frac{1}{\Gamma(\alpha) \Gamma(\beta)\theta^{\alpha + \beta}} (y_1y_2)^{\alpha - 1}(y_2 - y_1y_2)^{\beta - 1}e^{-y_2/\theta}
\end{align*}\) with support is \(0<y_1<1, 0<y_2<\infty\) It may be shown that the marginal pdf of \(Y_1\) is \(\begin{align*}
g(y_1) & = \frac{y_1^{\alpha - 1}(1 - y_1)^{\beta - 1}}{\Gamma(\alpha) \Gamma(\beta) } \int_0^{\infty} \frac{y_2^{\alpha + \beta -1}}{\theta^{\alpha + \beta}} e^{-y_2/\theta} dy_2
g(y_1) \\& = \frac{ \Gamma(\alpha + \beta) }{\Gamma(\alpha) \Gamma(\beta) } y_1^{\alpha - 1}(1 - y_1)^{\beta - 1}, \hspace{1cm} 0<y_1<1.
\end{align*}\) \(Y_1\) is said to have a beta pdf with parameters \(\alpha\) and \(\beta\). |
| Markdown | # 23\.2 - Beta Distribution
23\.2 - Beta Distribution
Let \\(X\_1\\) and \\(X\_2\\) have independent gamma distributions with parameters \\(\\alpha, \\theta\\) and \\(\\beta\\) respectively. Therefore, the joint pdf of \\(X\_1\\) and \\(X\_2\\) is given by
\\(\\begin{align\*} f(x\_1, x\_2) = \\frac{1}{\\Gamma(\\alpha) \\Gamma(\\beta)\\theta^{\\alpha + \\beta}} x\_1^{\\alpha-1}x\_2^{\\beta-1}\\text{ exp }\\left( -\\frac{x\_1 + x\_2}{\\theta} \\right), 0 \<x\_1 \<\\infty, 0 \<x\_2 \<\\infty. \\end{align\*}\\)
We make the following transformation:
\\(\\begin{align\*} Y\_1 = \\frac{X\_1}{X\_1+X\_2}, Y\_2 = X\_1+X\_2 \\end{align\*}\\)
The inverse transformation is given by
\\(\\begin{align\*} \&X\_1=Y\_1Y\_2, \\\\& X\_2=Y\_2-Y\_1Y\_2 \\end{align\*}\\)
The Jacobian is
\\(\\begin{align\*} \\left\| \\begin{array}{cc} y\_2 & y\_1 \\\\ -y\_2 & 1-y\_1 \\end{array} \\right\| = y\_2(1-y\_1) + y\_1y\_2 = y\_2 \\end{align\*}\\)
The joint pdf \\(g(y\_1, y\_2)\\) is
\\(\\begin{align\*} g(y\_1, y\_2) = \|y\_2\| \\frac{1}{\\Gamma(\\alpha) \\Gamma(\\beta)\\theta^{\\alpha + \\beta}} (y\_1y\_2)^{\\alpha - 1}(y\_2 - y\_1y\_2)^{\\beta - 1}e^{-y\_2/\\theta} \\end{align\*}\\)
with support is \\(0\<y\_1\<1, 0\<y\_2\<\\infty\\)
It may be shown that the marginal pdf of \\(Y\_1\\) is
\\(\\begin{align\*} g(y\_1) & = \\frac{y\_1^{\\alpha - 1}(1 - y\_1)^{\\beta - 1}}{\\Gamma(\\alpha) \\Gamma(\\beta) } \\int\_0^{\\infty} \\frac{y\_2^{\\alpha + \\beta -1}}{\\theta^{\\alpha + \\beta}} e^{-y\_2/\\theta} dy\_2 g(y\_1) \\\\& = \\frac{ \\Gamma(\\alpha + \\beta) }{\\Gamma(\\alpha) \\Gamma(\\beta) } y\_1^{\\alpha - 1}(1 - y\_1)^{\\beta - 1}, \\hspace{1cm} 0\<y\_1\<1. \\end{align\*}\\)
\\(Y\_1\\) is said to have a **beta** pdf with parameters \\(\\alpha\\) and \\(\\beta\\).
***
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