Let Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}. Assume that each pair (Xi, Yi) has correlation Corr(Xi, Yi) = ρ, but that (Xi,Yi) and (Xj,Yj) are independent for all i ̸= j.(a) What is Cov(Xi,Yi) in terms of σx, σy and ρ?(b) Show that Cov(Xi,Y ̄) = (ρσxσy)/n, where Y ̄ is the average of the Yi(c) Determine Cov(X ̄,Y ̄).B2. Consider the random variables Xi and Yi from question B1 again.(a) Show that the sample covariance is an unbiased estimator of Cov(X1,Y1).Hint: consider the equality Xi − X ̄ = (Xi − μ) − (X ̄ − μ).(b) Can you conclude from the statement in part (a) that the sample correlation is anunbiased estimator of Corr(X1, Y1)? Justify your answer.

It is given that: varxi=σx2varyi=σy2 and xi and yi are independent and corrxi,yi=ρ

Let Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}. Assume that each pair (Xi, Yi) has correlation Corr(Xi, Yi) = ρ, but that (Xi,Yi) and (Xj,Yj) are independent for all i ̸= j. (a) What is Cov(Xi,Yi) in terms of σx, σy and ρ? (b) Show that Cov(Xi,Y ̄) = (ρσxσy)/n, where Y ̄ is the average of the Yi (c) Determine Cov(X ̄,Y ̄).

Let Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}. Assume that each pair (Xi, Yi) has correlation Corr(Xi, Yi) = ρ, but that (Xi,Yi) and (Xj,Yj) are independent for all i ̸= j. (a) What is Cov(Xi,Yi) in terms of σx, σy and ρ? (b) Show that Cov(Xi,Y ̄) = (ρσxσy)/n, where Y ̄ is the average of the Yi (c) Determine Cov(X ̄,Y ̄).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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