Question

8.4. 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: y 1 fX,Y(x,y) 0 1 x Figure P8.4 (a) Make fully labeled sketches of the densities fX(x) and fY|X(y | 1/3). (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^2 and the bias b associated with the estimator: e^2 = E [(?MMSE(X) - Y)^2], 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.

          8.4. 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:

y
1
fX,Y(x,y)
0
1
x
Figure P8.4

(a) Make fully labeled sketches of the densities fX(x) and fY|X(y | 1/3).
(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^2 and the bias b associated with the estimator:
e^2 = E [(?MMSE(X) - Y)^2], 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.

8.4. 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:

y
1
fX,Y(x,y)
0
1
x
Figure P8.4

(a) Make fully labeled sketches of the densities fX(x) and fY|X(y | 1/3).
(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^2 and the bias b associated with the estimator:
e^2 = E [(?MMSE(X) - Y)^2], 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.

Added by Justin K.

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