Let X1 and X2 have independent gamma distributions with parameters , and ,

Chapter 5, Problem 5.2-6

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Let \(X_{1}\) and \(X_{2}\) have independent gamma distributions with parameters \(\alpha, \theta \text { and } \beta, \theta\) 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.)

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