Probability

Let X be a random variable, and let
ck=E[Xk].

We assume that X is such that ck exists (and is finite) for all k∈{0,1,2,⋯}. Define
Yk=Xk.

Please express your answers in terms of X,c1,c2,…,ck,… (for c1,c2,... please enter c1,c2,..., for ck please enter ck, and for ck+1,ck+2,... please enter ck1,ck2,...).

Find the Least Mean Squares (LMS) estimator of Yk based on X, as a function of X.

Y^kLMS= unanswered

Find the Linear Least Mean Squares (LLMS) estimator of Yk based on X.

Y^kLLMS= unanswered

Find the LMS estimator of Yk based on X3 (instead of X).

Y^kLMS= unanswered

Find the LLMS estimator of Yk based on X3 (instead of X).

Y^kLLMS= unanswered

For the remainder of the problem, we consider the LMS estimator of Yk based on X2 (instead of X). Under which of the conditions below would the LMS estimator of Yk based on X2 be equal to the LMS estimator of Yk based on X?

If k is even:

(i): They will never be equal.

(ii): They will always be equal.

(iii): They will not always be equal, but if X≥0 (i.e., P(X≥0)=1), then they are equal.

(iv): They will not always be equal, but if the distribution of X to be symmetric about 0, then they are equal.

(i)
(ii)
(iii)
(iv)
(iii)(iv)
unanswered

If k is odd:

(i): They will never be equal.

(ii): They will always be equal.

(iii): They will not always be equal, but if X≥0 (i.e., P(X≥0)=1), then they are equal.

(iv): They will not always be equal, but if the distribution of X to be symmetric about 0, then they are equal.

(i)
(ii)
(iii)
(iv)
(iii)(iv)
unanswered

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