Let X and Y be two jointly Gaussian real random variables, each with zero mean,variance 1, and correlation coefficient ρ ∈ (0, 1). Let a, b ∈ R be such that a2 + b2 = 1, and defineW := aX + bY .a. Find values for a and b to maximize the variance of W . Hint: Use eigendecomposition.b. Does the optimal W (from part a) have a probability density function? If yes, derive it. Ifnot, explain why.
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variance 1, and
W := aX + bY .
a. Find values for a and b to maximize the variance of W . Hint: Use eigendecomposition.
b. Does the optimal W (from part a) have a
not, explain why.
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