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TA3 - ECO2121 Methods of Economic Statistics TA3 4 February...

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ECO2121: Methods of Economic Statistics TA3 4 February 2009 1. Conditional Probability Recall Pr{ A B } = Pr{ A } + Pr{ B } Pr{ A B } Pr{ A B } = 0 if A and B are mutually exclusive. Suppose not, Pr{ A B } can be calculated first by conditioning on either event, i.e. Pr{ A B } = Pr{ A | B } × Pr{ B } or Pr{ A B } = Pr{ B | A } × Pr{ A } rearranging, Pr{ A | B } = Pr{ A B } / Pr{ B } or Pr{ B | A } = Pr{ A B } / Pr{ A } Hence, Pr{ A | B } = Pr{ A } and Pr{ B | A } = Pr{ B } if A and B are independent, which gives the usual formula Pr{ A B } = Pr{ A } × Pr{ B } Example: the probability of getting tail in tossing a coin and 6 in throwing a dice, given the coin and the dice are both fair and independent can be calculated as Pr{tail | 6} × Pr{6} = Pr{tail} × Pr{6} = 1/2 × 1/6 = 1/12 Bayes Theorem: Given { B i } i=1, ,k is a partition, for any (observed) event A, = × × = = k j j j i i i i B B A B B A A A B A B 1 } Pr{ } | Pr{ } Pr{ } | Pr{ } Pr{ } Pr{ } | Pr{ I Interpretations: Examples: Suppose a student is chosen from a class of 40 of which 28 are female, it is
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