MIT6_042JF10_rec21_sol

MIT6_042JF10_rec21_sol - 6.042/18.062J Mathematics for...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science December 1, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 21 1 Conditional Expectation and Total Expectation There are conditional expectations, just as there are conditional probabilities. If R is a random variable and E is an event, then the conditional expectation Ex ( R | E ) is defined by: Ex ( R | E ) = R ( w ) Pr { w | E } w S For example, let R be the number that comes up on a roll of a fair die, and let E be the event that the number is even. Lets compute Ex ( R | E ), the expected value of a die roll, given that the result is even. Ex ( R | E ) = R ( w ) Pr { w | E } w { 1 ,..., 6 } 1 1 1 = 1 0 + 2 + 3 0 + 4 + 5 0 + 6 3 3 3 = 4 It helps to note that the conditional expectation, Ex ( R | E ) is simply the expectation of R with respect to the probability measure Pr E () defined in PSet 10. So its linear: Ex ( R 1 + R 2 | E ) = Ex ( R 1 | E ) + Ex ( R 2 | E ) . Conditional expectation is really useful for breaking down the calculation of an expecta- tion into cases. The breakdown is justified by an analogue to the Total Probability Theorem: Theorem 1 (Total Expectation) . Let E 1 ,...,E n be events that partition the sample space and all have nonzero probabilities. If R is a random variable, then: Ex ( R ) = Ex ( R | E 1 ) Pr { E 1 } + + Ex ( R | E n ) Pr { E n } For example, let R be the number that comes up on a fair die and E be the event that result is even, as before. Then E is the event that the result is odd.odd....
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MIT6_042JF10_rec21_sol - 6.042/18.062J Mathematics for...

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