Let X be an unobserved random variable with EX 0, Var(X) 5 Assume that we have observed Y 1 , Y 2 , and Y 3 given by where EW 1 EW 2 EW 3 0, Var(W 1 ) 2, Var(W 2 ) 5, and Var(W 3 ) 3 Assume that W 1 , W 2 , W 3 , and X are independent random variables Find the linear MMSE estimator of X, given Y 1 ...

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Let X be an unobserved random variable with EX = 0, Var(X) = 5. Assume that we

Question:

Let X be an unobserved random variable with EX = 0, Var(X) = 5. Assume that we have observed Y₁, Y₂, and Y₃ given by Y = 2X + W, Y = X + W, Y3 = X + 2W3, where EW₁ = EW₂ = EW₃ = 0, Var(W₁) = 2, Var(W₂) = 5, and Var(W₃) = 3. Assume that W₁, W₂, W₃, and X are independent random variables. Find the linear MMSE estimator of X, given Y₁, Y₂, and Y₃.