how to solve it?

Problem 7. (20 points) It is well known that a random vector X of length N is Gaussian with mean vector y and covariance matrix C if and only if its characteristic function xw) 4 E{exp(jw? x)} is given by ox(W) = exp (jw'u (iz", -*) Answer the following questions. (a) (5 points) Find the mean vector and the covariance matrix of Y = AX + b. (b) (5 points) Show that any affine transform of X is Gaussian. (c) (10 points: specialize and generalize) When u = 4, and C = C, find A and b in terms of 4,4,C1, and Cy such that the mean vector and the covariance matrix of Y in (a) are given by E{Y} = 4, and Cov{Y} = C2, where C and C2 are symmetric positive-definite matrices. (Hint. Use the existence of symmetric positive-definite matrices C and C such that C = cc), and C2 = cc.) Problem 7. (20 points) It is well known that a random vector X of length N is Gaussian with mean vector y and covariance matrix C if and only if its characteristic function xw) 4 E{exp(jw? x)} is given by ox(W) = exp (jw'u (iz", -*) Answer the following questions. (a) (5 points) Find the mean vector and the covariance matrix of Y = AX + b. (b) (5 points) Show that any affine transform of X is Gaussian. (c) (10 points: specialize and generalize) When u = 4, and C = C, find A and b in terms of 4,4,C1, and Cy such that the mean vector and the covariance matrix of Y in (a) are given by E{Y} = 4, and Cov{Y} = C2, where C and C2 are symmetric positive-definite matrices. (Hint. Use the existence of symmetric positive-definite matrices C and C such that C = cc), and C2 = cc.)