PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Cov(X 1 , X 2 ) = E { (X 1 − µ 1 )(X 2 − µ 2 ) }
∫ ∞ ∫ ∞
= (x 1 − µ 1 )(x 2 − µ 2 )f(x 1 , x 2 ) dx 1 dx 2
−∞ −∞
∫ ∞ ∫ ∞
= (x 1 − µ 1 )(x 2 − µ 2 )f X1 (x 1 )f X2 (x 2 ) dx 1 dx 2 (since X 1 , X 2 independent)
−∞ −∞
(∫ ∞
) (∫ ∞
)
= (x 1 − µ 1 )f X1 (x 1 ) dx 1 (x 2 − µ 2 )f X2 (x 2 ) dx 2
−∞
−∞
= 0 × 0
= 0.
Remark:
But the converse does NOT apply, in general!
That is, Cov(X 1 , X 2 ) = 0 ⇏ X 1 , X 2 independent.
Definition. 1.9.2
If X 1 , X 2 are RVs, we define the symbol E[X 1 |X 2 ] to be the expectation of X 1 calculated
with respect to the conditional distribution of X 1 |X 2 ,
i.e.,
Theorem. 1.9.5
E [ X 1 |X 2
]
=
⎧⎪ ⎨
⎪ ⎩
∑
x 1
x 1 P X1 |X 2
(x 1 |x 2 ) X 1 |X 2 discrete
∫ ∞
−∞
Provided the relevant moments exist,
x 1 f X1 |X 2
(x 1 |x 2 ) dx 1
E(X 1 ) = E X2
{
E(X1 |X 2 ) } ,
X 1 |X 2 continuous
Var(X 1 ) = E X2
{
Var(X1 |X 2 ) } + Var X2
{
E(X1 |X 2 ) } .
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