Suppose X1, X2 are i.i.d. N(0, 1) random variables. Define: Y1 = aX1 + bX2, and Y2 = X1

(a) Find a and b such that Y = (Y1, Y2) is a Gaussian vector of standard normal with mean 0 and standard deviation 1 marginal distribution, and with correlation > 0.

(b) Compute the conditional expectation of Y1 given X1 , namely E[Y1|X1]

(c) Compute the conditional expectation of Y 2 1 given X1 , namely E[Y 2 1 |X1]