# 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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