Assume that two random variables X and Y are jointly Gaussian with m_x = m_y = 1, σ²_x = σ²_y = 4.

(a) Making use of (6.194), write down an expression for the marginal pdfs of X and of Y.

(b) Write down an expression for the conditional pdf f _{X | Y} (x | y) by using the result of (a) and an expression for f_XY (x,y) written down from (6.189). Deduce that f _{Y | X} (y | x) has the same form with y replacing x.

(c) Put f _{X | Y} (x | y) into the form of a marginal Gaussian pdf. What is its mean and variance? (The mean will be a function of y.)