Let X and Y be jointly normal random variables with parameters E(X) = E(Y ) =...

Let X and Y be jointly normal random variables with parameters E(X) = E(Y ) = 0, Var(X) = Var(Y ) = 1, and Cor(X, Y ) = ρ. For some nonnegative constants a ≥ 0 and b ≥ 0 with a² + b² > 0 define W1 = aX + bY and W2 = bX + aY .

(a)Show that Cor(W1, W2) ≥ ρ
(b)Assume that ρ = 0.1. Are W1 and W2 independent or not? Why?
(c)Assume now that ρ = 0.5. Find a ≥ 0 and b ≥ 0 such that E(W2|W1 = 1) = Var(W2|W1 = 1) = 0.5.

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orchestra answered 1 year ago

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