PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Define the mean vector by:
⎛ ⎞
µ 1
µ 2
µ = E(X) = ⎜ ⎟
⎝ . ⎠
µ r
and the variance matrix (covariance matrix) by:
Σ = Var(X) = [σ ij ] i=1,...,r
j=1,...,r
Finally, the correlation matrix is defined by:
⎛
⎞
1 ρ 12 . . . ρ 1r
. . .
. . .
R = ⎜
⎟
⎝ .. . ρr−1,r
⎠
1
where ρ ij =
σ ij
√
σii
√
σjj
.
Now suppose that a = (a 1 , . . . , a r ) T is a vector of constants and observe that:
a T x = a 1 x 1 + a 2 x 2 + · · · + a r x r =
r∑
a i x i
i=1
Theorem. 1.9.8
Suppose X has E(X) = µ and Var(X) = Σ, and let a, b ∈ R r be fixed vectors. Then,
1. E(a T X) = a T µ
2. Var(a t X) = a T Σa
3. Cov(a T X, b T X) = a T Σb
Remark
It is easy to check that this is just a re-statement of Theorem 1.9.3 using matrix
notation.
Theorem. 1.9.9
Suppose X is a random vector with E(X) = µ, Var(X) = Σ, and let A p×r and b ∈ R p
be fixed.
If Y = AX + b, then
E(Y) = Aµ + b and Var(Y) = AΣA T .
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