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Suppose we are allowed to observe a random process Z

Suppose we are allowed to observe a random process Z (t) at two points in time, to and t1. Based on those observations we would like to estimate Z (t) at time t= t2 where t0 < t1 < t2. We can view this as a prediction problem. Let our estimator be a linear combination of the two observations,

Y (t2) = Ẑ (t2) = aZ (t0) + bZ (t1)

(a) Use the orthogonality principle to find the MMSE estimator.

(b) Suppose that RZZ (t) = cexp (– b |τ|) for positive constants b and c. Show that in this case, the sample at time t = t2 is not useful for predicting the value of the process at time t = t1 > t0 (given we have observed the process at time). In other words, show that a = 0.

Y (t2) = Ẑ (t2) = aZ (t0) + bZ (t1)

(a) Use the orthogonality principle to find the MMSE estimator.

(b) Suppose that RZZ (t) = cexp (– b |τ|) for positive constants b and c. Show that in this case, the sample at time t = t2 is not useful for predicting the value of the process at time t = t1 > t0 (given we have observed the process at time). In other words, show that a = 0.

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