ISyE 6739 — Test 3 Solutions — Summer 2011

413. TRUE or FALSE? P(F (4, 3) < F 0.975,4,3 ) = P(F (4, 3) < 1/F 0.025,4,3 ).Solution: FALSE. The F 0.975,4,3 ) = 1/F 0.025,3,4 (flip the d.f.).♦14. Suppose X 1 , . . . , X 7 ∼ Nor(3, 6), Y 1 , . . . , Y 7 ∼ Nor(−3, 2), and everything isindependent. Let S 2 X and S 2 Y denote the sample variances of the X i ’s and Y j ’s,respectively. Name the distribution (with parameter(s)) of S 2 X/S 2 Y .Solution: SX 2 ∼ σXχ 2 2 (6)/6 and SY2σY 2 = Var(Y j ) = 2. Therefore,∼ σ 2 Y χ 2 (6)/6, where σ 2 X = Var(X i ) = 6 andS 2 XS 2 Y∼ σ2 Xχ 2 (6)/6σ 2 Y χ 2 (6)/6∼ 3F (6, 6). ♦15. The t distribution with 1 degree of freedom is also known as thedistribution. (Fill in the blank.)Solution: Cauchy.♦16. Suppose X 1 , X 2 , X 3 are i.i.d. Exp(λ), and we observe X 1 = 1, X 2 = 7, and X 3 = 4.What is the sample variance of the X i ’s?Solution: ¯X = 4 and n = 3. Thus,S 2 =∑ ni=1X 2 i − n ¯X 2n − 1= 9. ♦17. Suppose X 1 , X 2 , X 3 are i.i.d. Nor(µ, σ 2 ), and we observe X 1 = 1, X 2 = 7, andX 3 = 4. What is the sample variance of the X i ’s?Solution: ¯X = 4 and n = 3. Thus,S 2 =∑ ni=1Xi 2 − n ¯X 2= 9. ♦n − 1