Example: Multinormal distribution X ~ N(1, 4) and Y ~ N(-1, 1). (X, Y) follows multivariate normal distribution and E[(X - Y)2] = 7. Given X - Y = 1, find the conditional distribution of X. Solution: We need the joint distribution of (X, X - Y). . E[X - Y] = 1 - (-1) = 2. . Var(X - Y) = E[(X - Y)2] - (E[X - Y])2 = 7 -4 =3. . Cov(X, X - Y) = Cov(X, X) - Cov(X, Y) = 4 - Cov(X, Y). . Var(X - Y) = Var(X) - 2 Cov(X, Y) + Var(Y) implies Cov(X, Y) = } [Var(X) + Var(Y) - Var(X - Y)] = (4+1-3) = 1. . Cov(X, X - Y) = 4 -1 =3. X 4 (X - Y) ~N 10