Question: Suppose that \(W_{1}\) and \(W_{2}\) are independent \(\chi^{2}\) -distributed random variables with \(v_{1}\) and \(v_{2}\) df, respectively. According to Definition $7.3$,

has an \(F\) distribution with \(v_{1}\) and \(v_{2}\) numerator and denominator degrees of freedom, respectively. Use the preceding structure of \(F\), the independence of \(W_{1}\) and \(W_{2}\), and the result summarized in Exercise 7.30b to show

1. \(\quad E(F)=v_{2} /\left(v_{2}-2\right)\), if \(v_{2}>2\)
2. \(\quad V(F)=\left[2 v_{2}^{2}\left(v_{1}+v_{2}-2\right)\right] /\left[v_{1}\left(v_{2}-2\right)^{2}\left(v_{2}-4\right)\right]\), if \(v_{2}>4\)

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