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EES315 6.2 u2.pdf - 6.2 Event-based Independence Plenty of...

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6.2Event-based IndependencePlenty of random things happen in the world all the time, most ofwhich have nothing to do with one another. If you toss a coin andI roll a dice, the probability that you get heads is 1/2 regardless ofthe outcome of my dice. Events that are unrelated to each otherin this way are calledindependent.Definition 6.33. Two eventsA,Bare called (statistically27)independentifP(AB) =P(A)P(B)(9)Notation:A|=BRead “AandBare independent” or “Ais independent ofBWe call (9) themultiplication rulefor probabilities.If two events are not independent, they aredependent. In-tuitively, if two events are dependent, the probability of onechanges with the knowledge of whether the other has oc-curred.6.34.Intuition: Again, here is how you should think about inde-pendent events: “If one event has occurred, the probability of theother does not change.”P(A|B) =P(A)andP(B|A) =P(B).(10)In other words, “the unconditional and the conditional probabili-ties are the same”. We can almost use (10) as the definitions forindependence. This is what we mentioned in 6.8. However, we use(9) instead because it (1) also works with events whose probabili-ties are zero and (2) also has clear symmetry in the expression (sothatA|=BandB|=Acan clearly be seen as the same). In fact,in 6.37, we show how (10) can be used to define independence withextra condition that deals with the case when zero probability isinvolved.27Sometimes our definition for independence above does not agree with the everyday-language use of the word “independence”. Hence, many authors use the term “statisticallyindependence” to distinguish it from other definitions.80
Example 6.35.[25, Ex. 5.4] Which of the following pairs of eventsare independent?(a) The card is a club, and the card is black.Example: Club & Black1spadesclubsheartsdiamondsFigure 16: A Deck of Cards(b) The card is a king, and the card is black.6.36.An event with probability 0 or 1 is independent of any event(including itself).In particular,and Ω are independent of any events.One can also show that an eventAis independent of itself ifand only ifP(A) is 0 or 1.6.37.Now that we have 6.36, we can now extend the “practivaldefinition” from 6.34 to include events with zero probabilities:Two eventsA,Bwith positive probabilitiesare independent ifand only ifP(B|A) =P(B), which is equivalent toP(A|B) =P(A).WhenAand/orBhas zero probability,AandBare automat-ically independent.

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Probability theory, probabilities P

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