764 CHAPTER 28. THE NORMAL DISTRIBUTION

Proof: From Theorem 28.1.2 Σ = E((X−m)(X−m)∗

). Then by assumption,

Σ =

(σ2

1 00 Σp−1

). (28.3)

I need to verify that if E ∈ σ (X1) and F ∈ σ (X2, · · · ,Xp) , then P(E ∩F) = P(E)P(F) .

Let E = X−11 (A) and F = (X2, · · · ,Xp)

−1 (B) where A and B are Borel sets in R andRp−1 respectively. Thus I need to verify that

P([(X1,(X2, · · · ,Xp)) ∈ (A,B)]) =

µ(X1,(X2,··· ,Xp)) (A×B) = µX1(A)µ(X2,··· ,Xp) (B) . (28.4)

Using 28.3, Fubini’s theorem, and definitions,

µ(X1,(X2,··· ,Xp)) (A×B) =∫Rp

XA×B (x)1

(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)dx

=∫R

XA (x1)∫Rp−1

XB (X2, · · · ,Xp) ·

1

(2π)(p−1)/2√2π(σ2

1

)1/2 det(Σp−1)1/2

e−(x1−m1)

2

2σ21 ·

e−12 (x

′−m′)∗Σ−1p−1

(x′−m′

)dx′dx1

where x′ = (x2, · · · ,xp) andm′ = (m2, · · · ,mp) . Now this equals

∫R

XA (x1)1√

2πσ21

e−(x1−m1)

2

2σ21 ·

∫B

1

(2π)(p−1)/2 det(Σp−1)1/2 e

−12 (x

′−m′)∗Σ−1p−1

(x′−m′

)dx′dx. (28.5)

In case B = Rp−1, the inside integral equals 1 and

µX1(A) = µ(X1,(X2,··· ,Xp))

(A×Rp−1)= ∫

RXA (x1)

1√2πσ2

1

e−(x1−m1)

2

2σ21 dx1

which shows X1 is normally distributed as claimed. Similarly, letting A = R,

µ(X2,··· ,Xp) (B) = µ(X1,(X2,··· ,Xp)) (R×B)

=∫

B

1

(2π)(p−1)/2 det(Σp−1)1/2 e

−12 (x

′−m′)∗Σ−1p−1

(x′−m′

)dx′

and (X2, · · · ,Xp) is also normally distributed with mean m′ and covariance Σp−1. Nowfrom 28.5, 28.4 follows. In case the covariance matrix is diagonal, the above reasoningextends in an obvious way to prove the random variables,

{X1, · · · ,Xp

}are independent.