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44 Joint Gaussian Probability Density Function The joint Gaussian probability density function (pdf) of random variables X and Y is given by 29 fxY (1, y) = exp 8 + 18 (a) Find the marginals fx(x) and fy(y). (b) Are the random variables X and Y statistically dependent or independent? Explain. (c) Sketch the joint pdf fxy(x, y) on the square grid shown in Figure 6: 1) use a top view plot and contour lines to represent the amplitude of the joint pdf; and 2) indicate both the horizontal and vertical scales (location and spread of the joint pdf).\fX1, X2 and X3 are three Gaussian random variables with mean vector m and covariance matrix K given by 4 3 1 mx = KX = 3 5 2 1 2 Find the joint pdf of random variables Z1 and Z2 given by Z1 = 3X1 -2X2 + X3 Z2 = X1 + 3X2 - 2X3.5. (Marginally Gaussian but jointly non-Gaussian RVs) [10] Consider two jointly distributed RVs X and Y. If they are jointly Gaussian, by definition we know that X and Y are marginally Gaussian as well. However, the converse is not true. Here is a counter example. Let X ~ N(0, 1) and Y = ZX where Z ~ Unif({+1}), Z IL X. a) Show that Y ~ N(0, 1). CO b) Show that X + Y is not Gaussian. Hence, X and Y are not jointly Gaussian. c) Show that X and Y are uncorrelated but not independent