Let X1 and X2 have independent gamma distributions with

QUESTION:

Let \(X_{1}\) and \(X_{2}\) have independent gamma distributions with parameters \(\alpha, \theta\) and \(\beta, \theta\), respectively. Let \(W=X_{1} /\left(X_{1}+X_{2}\right)\). Use a method similar to that given in the derivation of the \(F\) distribution (Example 5.2-4) to show that the pdf of \(W\) is

\(g(w)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} w^{\alpha-1}(1-w)^{\beta-1}, \quad 0<w<1\).

We say that W has a beta distribution with parameters  \(\alpha\) and \(\beta\). (See Example 5.2-3.)

Equation Transcription:

Text Transcription:

X_1

X_2

Alpha, theta

Beta, theta

W=X_1/(X_1+X_2)

F  

W  

g(w)=gamma(alpha+beta)/gamma (alpha) gamma(beta)w^-1(1-w)^ beta-1, 0<w<1

Alpha

beta