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Review-Questions

# Review-Questions - ARE 210 Review Questions page 1 1 You...

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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 , 1 2 3 , , , , n x x x x " 1 2 3 , , , , n y y y y " The sample means and variances for X and Y , and the sample covariance between X and Y are 1 1 n i n i x x = = , 1 1 n i n i y y = = , 1 2 2 1 ( ) n x i n i s x x = = , 1 2 2 1 ( ) n y i n i s y y = = , 1 1 ( ) ( ) n xy i i n i s x x y y = = . The sum of these two variables, , 1,2,3,..., i i i z x y i n = + = , is of main interest in the ex- periment. Define the sample mean for the ' i z s , 1 1 n i n i z z = = . a. Prove that z x y = + . Define the sample variance for the ' i z s , 1 2 2 1 ( ) n z i n i s z z = = . b. Prove that 2 2 2 2 z x y xy s s s s = + + . c. Prove that ( ) ( ) ( ) E Z E X E Y = + . d. Prove that ( ) 1 2 2 ( ) ( ) ( ) 2 ( , ) 2 x y xy n V Z V X V Y Cov X Y = + + = σ + σ + σ e. Prove that if X and Y are independent, then ( , ) 0 Cov 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.

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ARE 210 Review Questions page 2 2. An experiment has sample space
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