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cs336f1018 - Whatwellcover Probability Basics Lecture 19...

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11/8/10 1 Probability Basics Lecture 19 What we’ll cover Tying up loose ends Bayes’ Theorem Random Variables Expectation Bounds: Bonferonni, Markov, Chebyshev’s Inequalities Inclusion Exclusion Principle The inclusion–exclusion principle states that if A and B are two (finite) sets, then Similarly, for three sets A , B and C , For the general case of the principle, let A 1 , ..., A n be finite sets. Then Example: Modeling Energy Saving Transmissions in Wireless Networks One node transmits on a random k out of N timeslots, the second node listens on a random k out of N timeslots. They are idle in the remaining N‐k timeslots. What is the probability that 2 nd node hears the first? Michael J. McGlynn, Steven A. Borbash. Birthday Protocols for Low Energy Deployment and Flexible Neighbor Discovery in Ad Hoc Wireless Networks (MobiHoc, 2001) The Key to Bayesian Methods P(A B) P(A|B) P(B) P(B|A) = ----------- = --------------- P(A) P(A) This is Bayes Rule Bayes, Thomas (1763) An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370-418 General Forms of Bayes Rule
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11/8/10 2 General Forms of Bayes Rule Based on Andrew L. Moore tutorials An Example Definition of Independent Events Events E and F are called independent iff P(E F)= P(E) P(F). Note that this is equivalent to P(E|F)=P(E) and P(F|E)=P(F). Random Variables A random variable is a function that maps a set of outcomes to real numbers (or subsets of real numbers). We use capital letters to represent random variables (Eg. X) and lower case letters to represent particular outcomes.
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