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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 shuf±ed 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 ²rst die, 6 on second Roulette: 13 is a possible outcome. Flip a coin ²ve times: H,T,T,H,T is a possible outcome Census: draw census tract CT34021 (Mercer County, NJ)
ORF 245 - Prof. Rigollet Fall 2012 Modeling Some experiments may not be really random. For example, the height of the ±ood 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 in±nite number of repetitions. Formalized by the law of large numbers As the number of observations goes to inFnity, the proportion of occurrences of a given outcome converges to the probability of this outcome. Example: ²ipping a coin. 1 100 10000 0.0 0.2 0.4 0.6 0.8 1.0 number of tosses proportions
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
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 ±rst 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 ±rst}={(H,T), (H,H)} {no heads}={(T,T)} We want to ±nd rules to compute the probability of events from the probability of outcomes.
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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Chapter2_large - ORF 245 Prof Rigollet Fall 2012 Chapter 2...

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