PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Remarks
1. Observe that:
P X1 (x 1 ) = ∑ x 2
∑
· · · ∑
x 3
x r
P (x 1 , . . . , x r )
= p 1 (x 1 ) ∑ ∑
· · · ∑
x 2
x 3
x r
p 2 (x 2 ) . . . p r (x r )
= c 1 p 1 (x 1 ).
Hence p 1 (x 1 ) ∝
↓
P X1 (x 1 )
} {{ } also p i(x i ) ∝ P Xi (x i )
sum of probs. marginal prob.
P X1 |X 2 ...X r
(x 1 |x 2 , . . . , x r ) =
=
=
=
=
P (x 1 , . . . , x r )
P x2 ...x r
(x 1 , . . . , x r )
p 1 (x 1 )p 2 (x 2 ) . . . p r (x r )
∑
x 1
p 1 (x 1 )p 2 (x 2 ) . . . p r (x r )
p 1 (x 1 )p 2 (x 2 ) . . . p r (x r )
p 2 (x 2 )p 3 (x 3 ) . . . p r (x r ) ∑ x 1
p 1 (x 1 )
p 1 (x 1 )
∑
x 1
p 1 (x 1 )
1
c P X 1
(x 1 )
1 ∑
P X1 (x 1 )
c
x 1
= P X1 (x 1 ).
That is, P X1 |X 2 ...X r
(x 1 |x 2 , . . . , x r ) = P X1 (x 1 ).
2. Clearly independence =⇒
P Xi |X 1 ...X i−1 ,X i+1 ...X r
(x i |x 1 , . . . , x i−1 , x i+1 , . . . , x r ) = P Xi (x i ).
Moreover, we have:
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