Estimation based on a function of the observation. Let be a positive random variable, with

known mean and variance 2, to be estimated on the basis of a measurement X of the

form X = and known fourth moment E[W4]. Thus, the conditional mean and variance of X given are 0 and , respectively, so we are essentially trying to estimate the variance of X given an observed value.

(a) Find the linear LMS estimator of based on X = x. (b) Let Y = X2. Find the linear LMS estimator of based on Y = y.

Estimation based on a function of the observation. Let E) be a positive random variable, with known mean ,a and variance 02, to be estimated on the basis of a measurement X of the form X : x/E_)W. We assume that W is independent of 9 with zero mean, unit variance, and known fourth moment E[W4]. Thus, the conditional mean and variance of X given 6) are 0 and 8, respectively, so we are essentially trying to estimate the variance of X given an observed value. (a) Find the linear LMS estimator of 8 based on X : on. (b) Let Y = X 2. Find the linear LMS estimator of 9 based on Y = y