PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Definition. 1.9.1.
If X 1 , X 2 are RVs with E[X i ] = µ i and Var(X i ) = σ 2 i , i = 1, 2 we define
σ 12 = Cov(X 1 , X 2 ) = E [ (X 1 − µ 1 )(X 2 − µ 2 ) ]
= E [ X 1 X 2
]
− µ1 µ 2 ,
Remark:
ρ 12 = Corr(X 1 , X 2 ) = σ 12
σ 1 σ 2
.
Cov(X, X) = Var(X) = { E[X 2 ] − (E[X]) 2} .
In some contexts, it is convenient to use the notation
Theorem. 1.9.3
σ ii = Var(X i ) instead of σ 2 i .
Suppose X 1 , X 2 , . . . , X r are RVs with E[X i ] = µ i , Cov(X i , X j ) = σ ij , and let a 1 , a 2 , . . . , a r ,
b 1 , b 2 , . . . , b r be constants. Then
( r∑
Cov a i X i ,
i=1
)
r∑
b j X j =
j=1
r∑ r∑
a i b j σ ij .
i=1 j=1
46