6 The Multivariate Normal Distribution The multivariate normal distribution with mean R d and positive definite covariance matrix R dd , denoted N(, ), has the probability density function f(x; , ) = 1 p (2) d || exp 1 2 (x ) 1 (x ) ! . Here || denotes the determinant of . You may use the following facts without proof. The volume under the normal PDF is 1. Z Rd f(x) dx = Z Rd 1 p (2) d || exp ( 1 2 (x ) 1 (x ) ) dx = 1. The change-of-variables formula for integrals: let f be a smooth function from R d R, let A R dd be an invertible matrix, and let b R d be a vector. Then, performing the change of variables x 7 z = Ax b, Z Rd f(x) dx = Z Rd f(A 1 z A 1 b) |A 1 | dz. 1. Let X N(, ). Use a suitable change of variables to show that E[X] = . 2. Use a suitable change of variables to