Solution Found!
Suppose that W1 and W2 are independent ? 2-distributed
Chapter 7, Problem 34E(choose chapter or problem)
Suppose that \(\mathrm{W}_{1} \text { and } \mathrm{W}_{2}\) are independent \(x^{2}\) -distributed random variables with \(\mathrm{v}_{1} \text { and } \mathrm{v}_{2}\) df, respectively. According to Definition 7.3,
\(\mathrm{F}=\frac{W_{1} / v_{1}}{W_{2} / v_{2}}\)
has an F distribution with \(\mathrm{v}_{1} \text { and } \mathrm{v}_{2}\) numerator and denominator degrees of freedom, respectively. Use the preceding structure of F, the independence of \(\mathrm{W}_{1} \text { and } \mathrm{W}_{2}\), and the result summarized in Exercise 7.30(b) to show
\(\mathrm{E}(\mathrm{F})=\mathrm{v}_{2} /\left(\mathrm{v}_{2}-2\right) \text {, if } \mathrm{v}_{2}>2\)\(\mathrm{V}(\mathrm{F})=\left[2 v_{2}^{2}\left(\mathrm{v}_{1}+\mathrm{v}_{2}-2\right)\right]\left[\left[\mathrm{v}_{1}\left(\mathrm{v}_{2}-2\right)^{2}\left(\mathrm{v}_{2}-4\right)\right], \text { if } \mathrm{v}_{2}>4\right.\)
Equation Transcription:
Text Transcription:
W1 and W2
v1 and v2
F=W1/v1W2/v2
v1 and v2
W1 and W2
E(F)=v2/(v2-2), if v2 >2
V(F)=[2 v22 (v1+v2-2)]/[v1(v2-4)], if v2 >4
Questions & Answers
QUESTION:
Suppose that \(\mathrm{W}_{1} \text { and } \mathrm{W}_{2}\) are independent \(x^{2}\) -distributed random variables with \(\mathrm{v}_{1} \text { and } \mathrm{v}_{2}\) df, respectively. According to Definition 7.3,
\(\mathrm{F}=\frac{W_{1} / v_{1}}{W_{2} / v_{2}}\)
has an F distribution with \(\mathrm{v}_{1} \text { and } \mathrm{v}_{2}\) numerator and denominator degrees of freedom, respectively. Use the preceding structure of F, the independence of \(\mathrm{W}_{1} \text { and } \mathrm{W}_{2}\), and the result summarized in Exercise 7.30(b) to show
\(\mathrm{E}(\mathrm{F})=\mathrm{v}_{2} /\left(\mathrm{v}_{2}-2\right) \text {, if } \mathrm{v}_{2}>2\)\(\mathrm{V}(\mathrm{F})=\left[2 v_{2}^{2}\left(\mathrm{v}_{1}+\mathrm{v}_{2}-2\right)\right]\left[\left[\mathrm{v}_{1}\left(\mathrm{v}_{2}-2\right)^{2}\left(\mathrm{v}_{2}-4\right)\right], \text { if } \mathrm{v}_{2}>4\right.\)
Equation Transcription:
Text Transcription:
W1 and W2
v1 and v2
F=W1/v1W2/v2
v1 and v2
W1 and W2
E(F)=v2/(v2-2), if v2 >2
V(F)=[2 v22 (v1+v2-2)]/[v1(v2-4)], if v2 >4
ANSWER:
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