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cs440-lec4-probability

cs440-lec4-probability - CS 440 ECE 448 Introduction to...

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CS 440 / ECE 448 Introduction to Artificial Intelligence Spring 2008 Instructor: Eyal Amir

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Where we are in this class So far: propositional logic, matrix  algebra Propositional logic: propositions,  formulas, truth assignments, formula  evaluation, entailment, CNF Matrix algebra: vectors, matrices, +, *,  transpose, eigen vectors, eigen values,  rank, inverse
Today: Probability Random experiment, domain ( ),  events Example:   = {H,T} (coin) Example:   = {1,2,3,4,5,6} (die) Example:   =  N N  = {1,2,3,…} (e.g., time) Random Variables (RVs), Probability  Distributions (P) This class: Discrete, finite domains

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Random Variables: Examples Random Variables: Examples X takes the value True (T) with probability p  and False (F) with probability 1-p is a  random  variable  with distribution (p,1-p) Formally: dom(X)={T,F}; P(X=T)=p; P(X=F)=1-p If an urn contains balls having 3 possible  colors – red, yellow, and blue – the color of a  ball picked at random from the bag is a  random variable with 3 possible values The  (probability) distribution  of a random  variable X with n values x 1 , x 2 , …, x n  is:                  (p 1 , p 2 , …, p n with P(X=x i ) = p i  and  Σ i=1,…,n  p i  = 1
Expected Value Expected Value Random variable X with n values x 1 ,…,x n   and distribution (p 1 ,…,p n ) E.g.: X is the state reached after doing an

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