Discussion
Suppose that \(X_{K}\), K = 1, 2, 3, 4 are independent and identically distributed random variables such that \(X_{1}\) has exponential distribution with scale parameter 1. Then, which of the following statements is/are true ?
(I) \(\dfrac{X_{1} + X_{2}}{X_{3} + X_{4}}\) has a F-distribution
(II) \(\dfrac{X_{1} + X_{2}}{X_{1} + X_{2} + X_{3}}\) has a \(\beta\)-distribution
Answer: A
Both (I) and (II) are true.
Each \(X_{k}\) is exponential with mean 1, that is Gamma(1, 1), and \(2X_{k} \sim \chi^{2}_{2}\).
(I) \(X_{1} + X_{2}\) and \(X_{3} + X_{4}\) are independent Gamma(2, 1), so \(2(X_{1} + X_{2})\) and \(2(X_{3} + X_{4})\) are independent \(\chi^{2}_{4}\). Their ratio, each divided by its 4 degrees of freedom, is
\[ \frac{X_{1} + X_{2}}{X_{3} + X_{4}} \sim F_{4, 4} \]
(II) For independent \(U \sim \text{Gamma}(a)\) and \(V \sim \text{Gamma}(b)\), \(\frac{U}{U + V} \sim \text{Beta}(a, b)\). With \(U = X_{1} + X_{2}\) (shape 2) and \(V = X_{3}\) (shape 1), the ratio is Beta(2, 1).
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