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

To find the Least Mean Squares (LMS) estimator of Yk based on X, we need to minimize the mean square error between the estimator and the true value of Yk. The LMS estimator is given by:

Y^kLMS = E[(X - Yk)^2 | X]

This can be calculated by minimizing the expectation of the squared difference between X and Yk, conditioned on X. The result will be a function of X, denoted as Y^kLMS.

To find the Linear Least Mean Squares (LLMS) estimator of Yk based on X, we use a linear combination of X and its powers to estimate Yk. The LLMS estimator is given by:

Y^kLLMS = a*X + b*X^2 + ... + c*X^k

Here, a, b, ..., c are constants determined by minimizing the mean square error. The resulting estimator Y^kLLMS is also a function of X.

To find the LMS estimator of Yk based on X^3, we substitute X^3 into the LMS formula and minimize the mean square error:

Y^kLMS = E[(X^3 - Yk)^2 | X^3]

Similarly, to find the LLMS estimator of Yk based on X^3, we substitute X^3 into the LLMS formula:

Y^kLLMS = a*X^3 + b*X^6 + ... + c*X^(3k)

Now, let's consider the situation where we use X^2 instead of X for the LMS estimator of Yk. Under which conditions would the LMS estimator of Yk based on X^2 be equal to the LMS estimator of Yk based on X?

If k is even, it means that k can be written as k = 2n, where n is an integer. In this case, the LMS estimator of Yk based on X^2 and X will always be equal. So the answer is (ii) They will always be equal.

If k is odd, it means that k can be written as k = 2n + 1, where n is an integer. In this case, the LMS estimator of Yk based on X^2 and X will not always be equal. However, if the distribution of X is such that X ≥ 0 (i.e., P(X ≥ 0) = 1), then the LMS estimators will be equal. So the answer is (iii) They will not always be equal, but if X ≥ 0, then they are equal.

If the distribution of X is symmetric about 0, then the LMS estimators of Yk based on X^2 and X will also be equal for odd values of k. So the answer is (iv) They will not always be equal, but if the distribution of X is symmetric about 0, then they are equal.

Therefore, the correct answers are (ii) and (iv).