Question: Supposeis a random process with mean function Y ( t ) = t and autocorrelation function R Y ( t , s ) = 2

Supposeis a random process with mean functionY(t)=t and autocorrelation functionRY(t,s)=2min(t,s)+2ts

fort0 ands0 and where>0 and2>0 are constants.

(a) Find the linear minimum mean square estimator (MMSE) ofYt+ based on the observation of {Y ; ot }. Also find the corresponding mean square error. Note is a known parameter.

(b) Suppose now that{Y_t; t0} is a homogeneous Poisson counting process with rate. Show that the estimator from part (a) with= is acutually the overall MMSE estimate ofYt+ (over all estimates, including non-linear ones).

Please also see the image:

Supposeis a random process with mean functionY(t)=t and autocorrelation functionRY(t,s)=2min(t,s)+2ts fort0 ands0

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