Theory of Statistics - George Mason University

2.11 The Family of Normal Distributions 189
1
(2π) nm/2 |Ψ| n/2 |Σ| m/2 e−tr(Ψ −1 (X−M) T Σ −1 (X−M))/2 ,
where M ∈ IR n×m , Ψ ≻ 0 ∈ IR m×m , and Σ ≻ 0 ∈ IR n×n .
The variance-covariance matrix of X is V(X) = V(vec(X)) = Ψ ⊗ Σ. The
variance-covariance matrix of each row of X is Ψ, and the variance-covariance
matrix of each column of X is Σ.
The multivariate matrix normal distribution of the matrix X with PDF
as given above is related to the ordinary multivariate normal for the vector
vec(X) with PDF
1
(2π) nm/2 |Ψ ⊗ Σ| nm/2 e−vec(X−M)T (Ψ⊗Σ) −1 vec(X−M)/2 .
Complex Multivariate Normal Distribution
Consider the random d-vector Z, where
Z = X + iY.
The vector Z has a complex d-variate normal distribution if (X, Y ) has a real
2d-variate normal distribution. The PDF of Z has the form
where µ ∈ IC d and Σ ≻ 0 ∈ IC d×d .
1
(2π) d/2 |Σ| 1/2e−(x−µ)H Σ −1 (x−µ)/2 ,
2.11.2 Functions of Normal Random Variables
One reason that the normal distribution is useful is that the distributions of
certain functions of normal random variables are easy to derive and they have
nice properties. These distributions can often be worked out from the CF of
the normal distribution N(µ, σ 2 ), which has a particularly simple form:
ϕ(t) = e iµt−σ2 t 2 /2 .
Given n iid N(µ, σ2 ) random variables, X1, X2, . . ., Xn, the sample mean
and sample variance
n
X = Xi/n (2.31)
and
S 2 =
are important functions.
n
Xi −
i=1
i=1
n
Xi/n
i=1
2
Theory of Statistics c○2000–2013 James E. Gentle
/(n − 1) (2.32)