You would like to know a sample value of X, a Gaussian (0,4) random variable. However, you only can observe noisy observations of the form Yi = X + N{. In terms of a vector of noisy observations, you observe

where N1 is a Gaussian (0,1) random variable and N2 is a Gaussian (0,2) random variable. Under the assumption that X, N1, and N2 are mutually independent, answer the following questions:
(a) Suppose you use Y1 as an estimate of X. The error in the estimate is D1 = Y1 - X. What are the expected error E[D1] and the expected squared error E[D21]?
(b) Suppose we use Y3 = (Y1 + Y2)/2 as an estimate of X. The error for this estimate is D3 = Y3 - X. Find the expected squared error E[D23]. Is Y3 or Y1 a better estimate of X?
(c) Let Y4 = AY where A = [a 1 - a] is a 1 × 2 matrix. Let D4 = Y4 - X denote the error in using Y4 as an estimate for X. In terms of a, what is the expected squared error E[D24]? What value of a minimize E[D24]?

Transcribed Image Text:

[Yi] Y2 [N [N2 +x H