Class2Notes

Class2Notes - 0.1. GENERAL CONSEQUENCES OF PROBABILITY...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
0.1. GENERAL CONSEQUENCES OF PROBABILITY AXIOMS 1 0.1 General Consequences of Probability Axioms Theorem 1 P [ A ] 1 ; 8 A 2 S Proof. A + A = S = ) P A + A ± = P [ S ] = 1 A A = = ) P A + A ± = P [ A ] + P A ± ) P [ A ] + P A ± = 1 = ) P [ A ] = 1 ± P A ± Note: P A ± ² 0 for any event A , by positivity axiom. ) P [ A ] 1 : Theorem 2 P [ A ] P [ B ] and P [ B ± A ] = P [ B ] ± P [ A ] if A ³ B; 8 A;B 2 S Proof. (Analogous to previous proof, with set B here playing role of S .) A + ( B ± A ) = B = ) P [ A + ( B ± A )] = P [ B ] A ( B ± A ) = = ) P [ A + ( B ± A )] = P [ A ] + P [ B ± A ] ) P [ A ] + P [ B ± A ] = P [ B ] = ) P [ A ] = P [ B ] ± P [ B ± A ] Note: P [ B ± A ] ² 0 for any event B ± A , by positivity axiom. ) P [ A ] P [ B ] Also: P [ A ] + P [ B ± A ] = P [ B ] = ) P [ B ± A ] = P [ B ] ± P [ A ] : Theorem 3 P [ A 1 [ A 2 [ ::: [ A Q ] = P [ A 1 ]+ P [ A 2 ]+ ::: + P [ A Q ] if A 1 ;A 2 ;:::;A Q 2 S Proof. By the additivity axiom: A 1 + A 2 + ::: + A Q = ( A 1 + A 2 + ::: + A Q 1 ) + A Q = ) P [ A 1 + A 2 + ::: + A Q ] = P [ A 1 + A 2 + ::: + A Q 1 ] + P [ A Q ] Likewise, P [ A 1 + A 2 + ::: + A Q 1 ] = P [ A 1 + A 2 + ::: + A Q 2 ] + P [ A Q 1 ] , so = ) P [ A 1 + A 2 + ::: + A Q ] = P [ A 1 + A 2 + ::: + A Q 2 ] + P [ A Q 1 ] + P [ A Q ] Continue in like manner by induction until the theorem statement is ob- tained. Theorem 4 P [ A 1 [ A 2 [ ::: [ A Q ] P [ A 1 ]+ P [ A 2 ]+ ::: + P [ A Q ] if A 1 ;A 2 ;:::;A Q 2 S are any (not necessarily disjoint) sets. Proof. First show that this holds for Q = 2 by partition of A 1 + A 2 : A 1 + A 2 = A 1 + A 1 A 2 where A 1 A 1 A 2 = . ) P [ A 1 + A 2 ] = P [ A 1 ] + P A 1 A 2 ± , by the additivity axiom. Note: A 1 A 2 ³ A 2 = ) P A 1 A 2 ± P [ A 2 ] , by Theorem 2. ) P [ A 1 + A 2 ] P [ A 1 ] + P [ A 2 ] Interpreting this last equation generically, it can be applied by induction to A 1 + A 2 + ::: + A Q = ( A 1 + A 2 + ::: + A Q 1 ) + A Q
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 = ) P [ A 1 + A 2 + ::: + A Q ] P [ A 1 + A 2 + ::: + A Q 1 ] + P [ A Q ] Likewise, P [ A 1 + A 2 + ::: + A Q 1 ] P [ A 1 + A 2 + ::: + A Q 2 ] + P [ A Q 1 ] , so P [ A 1 + A 2 + ::: + A Q ] P [ A 1 + A 2 + ::: + A Q 2 ] + P [ A Q 1 ] + P [ A Q ] Continue in like manner by induction until the theorem statement is ob- tained. (The process of induction here is analogous to that in the proof of Theorem 2, with " " here playing the role of " = " there.) Theorem 5 P [ A [ B ] = P [ A ] + P [ B ] ± P [ AB ] ; 8 A;B 2 S . Proof. P [ A + B ] = P [ A ] + P AB ± Note: AB = B ± AB ) P [ A + B ] = P [ A ] + P [ B ± AB ] Note: AB ² B = ) P [ B ± AB ] = P [ B ] ± P [ AB ] , by application of Theorem 2. Combining the last two results gives
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 18

Class2Notes - 0.1. GENERAL CONSEQUENCES OF PROBABILITY...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online