ARE 210 Review Questions page 1 1. You are given n observations from a random sample of an experiment on the random variables X and Y, 123,,,,nxxx x" 123,,,,nyyy y" The sample means and variances for X and Y, and the sample covariance between X and Y are 11ninixx==∑, 11niniyy==∑, 1221()nxinisxx==−∑, 1221()nyinisyy==−∑, 11()()nxy i inisxxyy==−⋅ −∑. The sum of these two variables, , 1, 2,3,...,iiizxyi n=+= , is of main interest in the ex-periment. Define the sample mean for the 'izs, 11ninizz==∑. a. Prove that zxy=+. Define the sample variance for the 'izs, 1221()nziniszz==−∑. b. Prove that 2222zxy xysss s=++ . c. Prove that () ( ) ()EZ EX EY=+. d. Prove that ()122() ( ) () 2 ( ,) 2xy xynVZ VX VY CovXY=++ =σ+σ+σ e. Prove that if X and Y are independent, then (,)0Cov X Y=. f. Is the converse to this true? Why or why not? If your answer is yes, prove it. If your answer is no, provide a counterexample.ARE 210 Review Questions page 2 2. An experiment has sample space S, an algebra of subsets of the sample space σ(S) = F, and a probability set function, P: F → [0,1]. For any three events A, B, C ∈ F prove that P(A B C) P(A) P(B) P(C) P(A B) P(A C) P(B C) P(A B C)∪∪= + + − ∩− ∩− ∩+ ∩∩. 3. For the sample space {}12345=S , construct the smallest σ-algebra that con-tains the singletons {1}, {2}, {3}, {4}, {5} as events. 4. Let ι = [1 1 ⋅⋅⋅ 1]′ be an n-dimensional column vector, with each element equaling one. a. Show that 1/()n−′′′=− =−AI Iιι ι ι ι ι is symmetric and idempotent. b. Show that the rank of A is n-1. c. Show that / n′xxA is the sample variance of the elements of the n-dimensional column vector x.. d. What is the vector Ax ? e. Show that 1()−′′=B ιιι ι is symmetric and idempotent. f. What is the vector Bx ? g. Show that AB = BA = 0, an n×n matrix of zeroes. 5. Prove that the diagonal elements of a symmetric, idempotent matrix must be in the in-terval from zero to one. 6. Prove that the eigen values of a symmetric, idempotent matrix each equals zero or one. 7. If the probability density function of X is 6(1 ) 0 1()0xxif xfxelsewhere−<<= find the probability density function of 3YX=. 8. Let 1,,nXX" be 2.. . ( , )iid n µσ . a. Find the best linear unbiased estimator for µ.ARE 210 Review Questions page 3 b. Find the log-likelihood function for (µ, σ2). c. Find the maximum likelihood estimators for (µ, σ2). d. Find the Cramer-Rao lower bound for the 2×2 variance-covariance matrix of any pair of unbiased estimators for (µ, σ2). e. Find the best estimators for (µ, σ2). f. Show that the best estimator for µ achieves the Cramer-Rao lower bound and that the best estimator for σ2 does not for any finite sample size n. g. Show that the maximum likelihood estimator for σ2 is biased and has a vari-ance that is less than the Cramer-Rao lower bound. h. Prove that the best estimator for µ is independent of the best estimator for σ2. i. What is the distribution of 11ninixx==∑? j. What is the distribution of 122(1)1()ninisxx−==−∑? k. What is the distribution of ()()nxs−µ ? l. What is the distribution of 22()()xsn−µ ? m. How would you construct a 95% confidence interval for µ with x and 2s ? n. For the null hypothesis 00:Hµ=µ and the alternative hypothesis 10:H µ≠µ with σ2 unknown, how would you construct a test of H0 against H1 at signifi-cance level α? What is the distribution of the test statistic?ARE 210 Review Questions page 4 o. Suppose H0 is false. What is the noncentrality parameter for the test statistic? How does the noncentrality parameter vary with the true value of µ and with the sample size n? How is this related to the power of the