Consider two random variables x, y whose joint PDF fxy (x, y) is shown in Figure...

   

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Consider two random variables x, y whose joint PDF fxy (x, y) is shown in Figure (a) Provide a mathematical expression for fxy(x, y). (b) Provide expressions for the marginal PDFs fx(x), fy(y) and sketch them. Can you deduce these marginal PDFs only graphically? Can you deduce whether x and y are statistically independent? Argue these points in your submission. (c) Calculate the first moments m), m) and the second moments m2), m). (d) Assume that we can only measure realizations of y, but cannot observe real- izations of x directly. Calculate the inhomogeneous linear MMSE estimator which gives an estimate x of x using the measured data y. (e) Calculate the corresponding MMSE of the estimator calculated in d). (f) Again, assume that we can only measure realizations of y, but cannot observe realizations of x directly. Calculate the nonlinear MMSE estimator which gives an estimate X of x using the measured data y. (g) Calculate the corresponding MMSE of the estimator calculated in f). constant height -3 -2 -1 -1 -2 -3 3 N 1 2 -N 2- 3 4 x Consider two random variables x, y whose joint PDF fxy (x, y) is shown in Figure (a) Provide a mathematical expression for fxy(x, y). (b) Provide expressions for the marginal PDFs fx(x), fy(y) and sketch them. Can you deduce these marginal PDFs only graphically? Can you deduce whether x and y are statistically independent? Argue these points in your submission. (c) Calculate the first moments m), m) and the second moments m2), m). (d) Assume that we can only measure realizations of y, but cannot observe real- izations of x directly. Calculate the inhomogeneous linear MMSE estimator which gives an estimate x of x using the measured data y. (e) Calculate the corresponding MMSE of the estimator calculated in d). (f) Again, assume that we can only measure realizations of y, but cannot observe realizations of x directly. Calculate the nonlinear MMSE estimator which gives an estimate X of x using the measured data y. (g) Calculate the corresponding MMSE of the estimator calculated in f). constant height -3 -2 -1 -1 -2 -3 3 N 1 2 -N 2- 3 4 x