Stick breaking. Given a stick of length 1, break it into two pieces at a location chosen uniform at random. Denote the breaking location by X, then X ? Unif[0,1]. Keep the piece corresponding to the interval [X, 1]. Break it again into two pieces at a location chosen uniform at random. Denote the second breaking location by Y , then Y |{X = x} ? Unif[x, 1].

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(a) Find the estimator of X given Y that minimizes the MSE E[(X? ? X)2]. (b) Find the conditional MSE given Y = y for the estimator you find in part (a).

(c) Find the covariance Cov(X, Y ). (d) Find the linear LMS estimator of X given Y .

(e) Find the MSE for the estimator you find in part (d)

Stick breaking. Given a stick of length 1, break it into two pieces at a location chosen uniform at random. Denote the breaking location by X, then X N Unif[0, 1]. Keep the piece corresponding to the interval [X , 1]. Break it again into two pieces at a location chosen uniform at random. Denote the second breaking location by Y, then Y|{X : 11:} ~ Unif[:1:, 1]. (a) Find the estimator of X given Y that minimizes the MSE E[(X - X) 2]. (b) Find the conditional MSE given Y = y for the estimator you find in part (a). (c) Find the covariance Cov(X, Y). (d) Find the linear LMS estimator of X given Y. (e) Find the MSE for the estimator you find in part (d)