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Chapter2_large - ORF 245 Prof Rigollet Fall 2012 Chapter 2...

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ORF 245 - Prof. Rigollet Fall 2012 Probability Chapter 2
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ORF 245 - Prof. Rigollet Fall 2012 A random experiment is an action or process whose outcome is uncertain. Random experiments Examples: Roll dice, draw cards from shuffled decks, picking a person at random for a survey, choosing a census tract at random An (elementary) outcome is one of the possible outcomes of a random experiment. Roll 2 dice: (2,6) is one possible outcome. 2 on first die, 6 on second Roulette: 13 is a possible outcome. Flip a coin five times: H,T,T,H,T is a possible outcome Census: draw census tract CT34021 (Mercer County, NJ)
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ORF 245 - Prof. Rigollet Fall 2012 Modeling Some experiments may not be really random. For example, the height of the flood in Holland depends on the moon, currents, temperature,... and many other parameters. It may still be helpful to consider it as a random experiment. This is called modeling (we use a simple model for the truth). Other examples are measurement errors or rounding unobservable characteristics of individuals (drug testing) stock market
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ORF 245 - Prof. Rigollet Fall 2012 Probability The probability of an outcome is the proportion of times the event would occur if we observed the random experiment for an infinite number of repetitions. Formalized by the law of large numbers As the number of observations goes to infinity, the proportion of occurrences of a given outcome converges to the probability of this outcome. Example: flipping a coin. 1 100 10000 0.0 0.2 0.4 0.6 0.8 1.0 number of tosses proportions
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ORF 245 - Prof. Rigollet Fall 2012 Flipping a coin coins=(runif(100000)<0.5) proportions=cumsum(coins)/(1:100000) plot(proportions, log="x", type="l", ylim=c(0,1), xlab="number of tosses") lines(1:100000, (1:100000)*0+0.5, col=2, lty=2) The proportion stabilizes around 1/2 1 100 10000 0.0 0.2 0.4 0.6 0.8 1.0 number of tosses proportions
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ORF 245 - Prof. Rigollet Fall 2012 Rolling a die face_one=(runif(100000)<(1/6)) proportions=cumsum(face_one)/(1:100000) plot(proportions, log="x", type="l", ylim=c(0,1), xlab="number of rolls") lines(1:100000, (1:100000)*0+1/6, col=2, lty=2) The proportion stabilizes around 1/6 1 100 10000 0.0 0.2 0.4 0.6 0.8 1.0 number of rolls proportions
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ORF 245 - Prof. Rigollet Fall 2012 We write P(outcome) the probability of an outcome. If all outcome are equally likely then Consider a more complicated example: rolling 2 dice. What is P({2,6})? It is the same as P({1,1}) or P({3,4}). All outcomes are equally likely. We need to count the outcomes or have rules to compute probabilities. Probability Rolling a die: P(2)=1/6 Flipping a coin: P(H)=1/2 P(outcome) = 1 number of outcomes
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ORF 245 - Prof. Rigollet Fall 2012 An event is a collection of outcomes. It can be described either with words or using formal notation from set theory. Passing from the first one to the second is a necessary skill. Events Flipping two coins. We know that the outcomes are (H,H), (H,T), (T, H), (T,T). Consider the events {twice the same}={(H,H), (T,T)} {heads first}={(H,T), (H,H)} {no heads}={(T,T)} We want to find rules to compute the probability of events from the probability of outcomes.
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ORF 245 - Prof. Rigollet Fall 2012 Operations on events A B
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ORF 245 - Prof. Rigollet Fall 2012 Union of two events A and B: Operations on events A B
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