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 \(\beta\).