LetXandYbe r.v.s with finite second moments, and set eX= μ₁, eY = μ₂, 0 < Var(X) = σ²₁, 0 < Var(Y) = σ²₂. Then covariance and the correlation coefficient ofXandY, denoted respectively, by Cov(X, Y) and ρ(X, Y), are defined by: Cov(X, Y) = e[(X €“ μ₁) (Y €“ μ₂)] = e(XY) €“ μ₁μ₂and ρ(X, Y) = Cov(X, Y)/σ₁σ₂.

(i) Then show that €“ σ₁ σ₂ £ Cov(X, Y) £ σ₁ σ₂, and Cov(X, Y) = σ₁ σ₂ if and only if

And Cov(X, Y) = 1 €“σ₁ σ₂ if and only if

(ii) Also, €“ 1 £ ρ(X, Y) £ 1, and ρ(X, Y) = 1 if and only if

And ρ(X, Y) = €“ 1 if and only if

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