# Question: Let and be jointly Gaussian random variables Show that Z

Let and be jointly Gaussian random variables. Show that Z = aX + bY is also a Gaussian random variable. Hence, any linear transformation of two Gaussian random variables produces a Gaussian random variable.

## Answer to relevant Questions

Let and be jointly Gaussian random variables with E [X] = 1, E [Y] = –2, Var (X) = 4, Var (Y) = 9, and ρX, Y. Find the PDF of Z = 2X – 3Y – 5. Let and be independent zero- mean, unit- variance Gaussian random variables. Consider forming the new random variable U, V according to U = [X] cos(θ) –[Y ] sin(θ) V = [X] sin (θ – [Y] cos (θ). Suppose M and N are independent discrete random variables with identical Poisson distributions, Find the PMF of L = M– N For positive constants a, b, c, and positive integer n, a pair of random variables has a joint PDF specified by (a) Find the constant in terms of a, b, c, and n. (b) Find the marginal PDFs, fX (x) and fY (y). (c) Find Pr ...Repeat parts (c) and (d) of Exercise 6.9 if a three element vector is formed from three rolls of a die. (c) Determine the mean vector; (d) Determine the covariance matrix.Post your question