Suppose that two random variables X and Y have a joint PDF fx,y(x, y) that is...

 

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Suppose that two random variables X and Y have a joint PDF fx,y(x, y) that is constant in the shaded region shown in Figure P8.4, and zero elsewhere: fxx(x,y) 0 1 Figure P8.4 (a) Make fully labeled sketches of the densities fx(x) and fyx(y). (b) Are X and Y statistically independent? Explain. (c) Determine and make a fully labeled sketch (as a function of x) of MMSE(X), the MMSE estimator of Y based on observing X. (d) To evaluate how well your estimator from (c) will perform on average, determine the mean square error e and the bias b associated with the estimator: =E[MMSE(X) - Y)], and b = E [MMSE(X) - Y], where the expectation is over X and Y jointly. (e) Determine LMMSE (X), the linear MMSE estimator of Y, and its associated MMSE. Suppose that two random variables X and Y have a joint PDF fx,y(x, y) that is constant in the shaded region shown in Figure P8.4, and zero elsewhere: fxx(x,y) 0 1 Figure P8.4 (a) Make fully labeled sketches of the densities fx(x) and fyx(y). (b) Are X and Y statistically independent? Explain. (c) Determine and make a fully labeled sketch (as a function of x) of MMSE(X), the MMSE estimator of Y based on observing X. (d) To evaluate how well your estimator from (c) will perform on average, determine the mean square error e and the bias b associated with the estimator: =E[MMSE(X) - Y)], and b = E [MMSE(X) - Y], where the expectation is over X and Y jointly. (e) Determine LMMSE (X), the linear MMSE estimator of Y, and its associated MMSE.