Page 268 - Elements of Distribution Theory

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254 Normal Distribution Theory

8.10 Let X denote a d-dimensional random vector with a multivariate normal distribution with mean
d
vector 0 and covariance matrix I d . Let {v 1 ,..., v d } be an orthonormal basis for R and let
Y 1 ,..., Y d denote real-valued random variables such that
X = Y 1 v 1 + ··· + Y d v d ;

then Y 1 ,..., Y d are the coordinates of X with respect to {v 1 ,..., v d }.
(a) Find an expression for Y j , j = 1,..., d.
(b) Find the distribution of (Y 1 ,..., Y d ).
8.11 Let X denote a d-dimensional multivariate normal random vector with mean vector µ and
2
2
d
covariance matrix σ I d where σ > 0. Let M denote a linear subspace of R such that µ ∈ M.
T
T
d
d
T
Let c ∈ R be a given vector and consider Var(b X) where b ∈ R satisfies b µ = c µ. Show
T
that Var(b X)is minimized by b = P M c.
8.12 Let X denote a d-dimensional random vector with a multivariate normal distribution with mean
d
vector µ and covariance matrix given by I d . Suppose that R may be written
d
R = M 1 ⊕ M 2 ⊕ ··· ⊕ M J
d
where M 1 , M 2 ,..., M J are orthogonal linear subspaces of R . Let P M j denote orthogonal
X, j = 1,..., d. Show that Y 1 ,..., Y J are independent.
projection onto M j and let Y j = P M j
8.13 Let X be a d-dimensional random vector with a multivariate normal distribution with mean
vector 0 and covariance matrix . Write X = (X 1 , X 2 ) where X 1 is p-dimensional and X 2 is
(d − p)-dimensional, and

11 12
=
21 22
where 11 is p × p, 12 = T 21 is p × (d − p), and 22 is (d − p) × (d − p); assume that
| 2 2| > 0.
T
Find E[X X 1 |X 2 = x 2 ].
1
8.14 Let X = (X 1 , X 2 , X 3 ) denote a three-dimensional random vector with a multivariate normal
distribution with mean vector 0 and covariance matrix . Assume that
Var(X 1 ) = Var(X 2 ) = Var(X 3 ) = 1
and let
ρ ij = Cov(X i , X j ), i = j.
(a) Find the conditional distribution of (X 1 , X 2 )given X 3 = x 3 .
(b) Find conditions on ρ 12 ,ρ 13 ,ρ 23 so that X 1 and X 2 are conditionally independent given
X 3 = x 3 .
(c) Suppose that any two of X 1 , X 2 , X 3 are conditionally independent given the other random
variable. Find the set of possible values of (ρ 12 ,ρ 13 ,ρ 23 ).
8.15 Let X denote a d-dimensional random vector and let A denote a d × d matrix that is not
T
T
symmetric. Show that there exists a symmetric matrix B such that X AX = X BX.
8.16 Let X denote a d-dimensional random vector with a multivariate normal distribution with mean
0 and covariance matrix I d . Let A 1 and A 2 be d × d nonnegative-definite, symmetric matrices
T
and let Q j = X A j X, j = 1, 2, and let r j denote the rank of A j , j = 1, 2. Show that Q 1 and
Q 2 are independent chi-squared random variables if and only if one of the following equivalent
conditions holds:
(i) A 1 A 1 = A 1 and A 2 A 2 = A 2
(ii) r 1 + r 2 = r
(iii) A 1 A 2 = A 2 A 1 = 0.