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Paul Munro -Introduction to Information Science -IS2000 Probability Review • • • • • Definitions/Identities Random Variables Expected Value Joint Distributions Conditional Probabilities Paul Munro -Introduction to Information Science -IS2000 Probability Defined • an event (experiment) has a set of possible outcomes, each with a probability, that measures their relative (anticipated) frequencies of occurrence normalized to 1. Paul Munro -Introduction to Information Science -IS2000 Probability Identities Events and outcomes: E {e1,e2, ,en } pi Pr(ei ) Probability distribution: (p1, p2, , pn ) Probability of each outcome: 0 pi 1 for all i n pi 1 i1 Paul Munro -Introduction to Information Science -IS2000 Joint Distributions • Two (or more) events • Each event has an outcome • Joint distribution stipulates the probability of every combination of outcomes IS2000 -- Introduction to Information Science -- Paul Munro Two Events E {e1, ,en } E F F { f1, , f m } {(e1, f1 ), ,(e1, f m ), (e2 , f1 ), ,(e2 , f m ), (en , f1 ), ,(en , f m )} IS2000 -- Introduction to Information Science -- Paul Munro Random Variables Random variables A rando m va riable X takes on a value from a given set. Thus, it is an event where the outcome s x1, x2, ..., x N have numerical values. N The expe cted value of X is pi x i . i 1 Example: Find the expected value of X if Pr(X = 2) = 0.15 Pr(X = 6) = 0.20 Pr(X = 5) = 0.45 Pr(X = 8) = 0.20 Answer: N pi x i i 1 (0.15)(2) (0.45)(5) (0.20)(6) (0.20)(8) 0.3 2.25 1.20 1.60 5.35 Paul Munro -Introduction to Information Science -IS2000 Multiple Random Variables More than one random variable Two random variables X and Y , with X {x1, x2, ..., x N} and Y {y1, y2, ..., y M} Let X and Y be two simultaneous events with outcomes xi and yj. This joint event has a probabili ty p(xi , yj ). These probabilit ies can be written in matrix form. Note that the row s sum to the total probabili ty of the correspondin g xi , and the column s sum to the total probabili ty of the correspondin g yj. p(x1 , y1 ) p(x 1 ,y 2 ) ... p(x 1 , y M ) p(x1 ) p(x 2 , y1 ) p(x 2 ,y 2 ) ... p(x 2 , y M ) p(x 2 ) ... ... ... ... p(x N ,y 1 ) p(x N , y 2 ) ... p(x N , y M ) p(y1 ) p(y 2 ) ... ... p(y M ) p(x N ) 1 IS2000 -- Introduction to Information Science -- Paul Munro Joint probability matrix p(x1 , y1 ) p(x 1 ,y 2 ) ... p(x 1 , y M ) p(x1 ) p(x 2 , y1 ) p(x 2 ,y 2 ) ... p(x 2 , y M ) p(x 2 ) ... ... ... ... p(x N ,y 1 ) p(x N , y 2 ) ... p(x N , y M ) p(y1 ) p(y 2 ) ... ... p(x N ) p(y M ) 1 The sums of the columns and rows are ma thematically expressed as follows : Rows: p(x i ) M N p(x i ,y j ) Columns: p(y j ) p(x i , y j ) j1 i1 The sum of all the joint probabili ties is 1: N M N M N M i 1 j 1 i1 j1i 1 j 1 p(x i , y j ) p(x i ) p(x i , y j ) p(y j ) 1 . Paul Munro -Introduction to Information Science -IS2000 Conditional Probability • Random variables are often NOT independent • P(rain in Pittsburgh), P(rain in Monroeville), P(rain in New York), P(rain in Hong Kong) • P(Heads up), P(Tails down) • P(D1=5), P(D2=6) • P(D1=1), P(D1 + D2=2) IS2000 -- Introduction to Information Science -- Paul Munro Dice Example IS2000 -- Introduction to Information Science -- Paul Munro Conditional Probability A p(x i | y j ) AB p(x i , y j ) p(y j ) OR B P(A|B) = P(AB) P(B) p(x i ,y j ) p(x i | y j )p(y j ) p(y j | x i ) p(x i ) p(x 1 , y1 ) 0.1 p(x 1 , y 2 ) 0.0 p(x1 ,y 3 ) 0.2 p(x 2 ,y 1 ) 0.1 p(x 2 , y 2 ) 0.1 p(x 2 ,y 3 ) 0.5 p(y1) = 0.2 p(y2) = 0.1 p(y3) = 0.7 p(x 1 | y1 ) 0.5 p(x1 |y 2 ) 0.0 p(x1 | y 3 ) p(x 2 | y1 ) 0.5 p(x 2 | y 2 ) 1.0 p(x 2 | y 3 ) IS2000 -- Introduction to Information Science -- Paul Munro Example 2 7 5 7 Paul Munro -Introduction to Information Science -IS2000 Markov Processes • State transition probabilities • Matrix or Diagram • Matrix Multiplication predicts multiple transition probabilities • Mk Converges to steady state