PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Remark
The same methods can be used to establish a similar result for block diagonal matrices.
The simplest case is the following, which is most easily proved using moment generating
functions:
X 1 , X 2 are independent
if and only if Σ 12 = 0, i.e.,
( ) (( ) ( ))
X1
µ1 Σ11 0
∼ N
X
r1 +r 2
,
.
2 µ 2 0 Σ 22
Theorem. 1.10.7
Suppose X 1 , X 2 , . . . , X n are i.i.d. N(µ, σ 2 ) RVs and let
¯X = 1 n
S 2 =
n∑
X i ,
i=1
1
n − 1
n∑
(X i − ¯X) 2 .
i=1
Then ¯X ∼ N(µ, σ 2 /n) and
(n − 1)S2
σ 2
∼ χ 2 n−1 independently.
(Note: S 2 here is a random variable, and is not to be confused with the sample covariance
matrix, which is also often denoted by S. Hopefully, the meaning of the notation
will be clear in the context in which it is used.)
Proof. Observe first that if X = (X 1 , . . . , X n ) T then the i.i.d. assumption may be
written as:
X ∼ N n (µ1 n , σ 2 I n×n )
1. ¯X ∼ N(µ, σ 2 /n):
Observe that ¯X = BX, where
[ 1
B =
n
]
1
n . . . 1
n
Hence, by Theorem 1.10.5,
¯X ∼ N(Bµ1, σ 2 BB T ) ≡ N(µ, σ 2 /n),
as required.
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