Consider again Problem 8, in which X is an unobserved random variable with EX = 0, Var(X) = 5. Assume that we have observed Y₁ and Y₂ given by

Y₁ = 2X +W₁,

Y₂ = X +W₂,

where EW₁ = EW₂ = 0, Var(W₁) = 2, and Var(W₂) = 5. Assume that W₁, W₂ , and X are independent random variables. Find the linear MMSE estimator of X, given Y₁ and Y₂, using the vector formula

Problem 8

Let X be an unobserved random variable with EX = 0, Var(X) = 5. Assume that we have observed Y₁ and Y₂ given by

Y₁ = 2X +W₁,

Y₂ = X +W₂,

where EW₁ = EW₂ = 0, Var(W₁) = 2, and Var(W₂) = 5. Assume that W₁, W₂ , and X are independent random variables. Find the linear MMSE estimator of X, given Y₁ and Y₂.

Transcribed Image Text:

XL = CXYCY ¹(Y - E[Y]) + E[X].