Discussion3

# Discussion3 - Discussion 3 Emily Mower February 1 2010...

This preview shows pages 1–4. Sign up to view the full content.

Discussion 3 Emily Mower February 1, 2010 Topics to be covered: Helpful probabilistic concepts Bayes Rule Partitioning the sample space 1 Helpful probabilistic concepts 1.1 Conditional probability The conditional probability of B, given A, denoted by P ( B | A ) is deﬁned by: P ( B | A ) = P ( A B ) P ( A ) , provided P ( A ) > 0 1.2 Multiplicative rules If in an experiment the events A and B can both occur, then: P ( A B ) = P ( A ) P ( B | A ) , provided P ( A ) > 0 This comes from multiplying both sides of the conditional probabilty deﬁni- tion by P ( A ). 1

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

View Full Document
2 Partitioning the sample space If the events A 1 , A 2 , ..., A k constitute a partition of the sample space S such that P ( A i ) 6 = 0 for i = 1 , 2 ,...,k , then for any event B of S , P ( B ) = k X i =1 P ( A i B ) = k X i =1 P ( A i ) P ( B | A i ) Note: The events A n ( n = { 1 , 2 ,...,k } ) are disjoint (see Figure 1)! Figure 1: Sample space partition (image source: http://www.math.uah.edu/stat/prob/Total.png) . 2
Bayes Rule If the events A 1 ,A 2 ,...A k constitute a partition of the sample space S such that P ( A i ) 6 = 0 for i = 1 , 2 ,...,k , then for any event B in S such that P ( B ) 6 = 0, P ( A r | B ) = P ( A r B ) k i =1 P ( A i B ) = P ( A r ) P ( B | A r ) k i =1 P ( A i ) P ( B | A i ) for r = 1 , 2 ,...,k Let’s look at the example for two events, A and B : P ( A | B ) = P ( A B ) P ( B ) P ( B | A ) = P ( A B ) P ( A ) This implies that: P ( A | B ) = P ( B | A ) P ( A ) P ( B ) The formulation above arrives at the same conclusion. Since P ( B ) can also

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 10

Discussion3 - Discussion 3 Emily Mower February 1 2010...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online