Problem 5.18 and 1

Problem 5.18 {plotting the joint Gaussian density] For jointly Gaussian random variables X and Y, plot the density and its contours as in Figure 5.15 for the following parameters: a a? :1?02:1? :0. Eblltrgv:l?arl,:05 3 -0.751 1. Let X = (X1, X2, X3) be jointly Gaussian with My = (1, -2, 3) and Cx = 3 4 0.25 0.75 0.25 1 Hint: If a set of random variables (RVs) are jointly Gaussian, then any subset of those RVs are also jointly Gaussian. Similarly, adding constants to (or taking linear combinations of) jointly Gaussian RVs results in jointly Gaussian RVs. Using this property you can solve problem 1 without using integration. When appropriate, you may express your answer by saying that the random vector or variable is Gaussian then specifying the mean and (co) variance. a) Find the variance for each X; (i.e., ox, , ox, , and ox, ). b) Find the correlation coefficients between X; and X; (i.e., (x,,x2, X1,X3, and rx2,X3). c) Find fx1,X3' d) Find fx3. e) Let Y = X - ux and find fy. f) Let Z = YA where the columns of A are the eigenvectors of Cy. Find fz