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class03_11

# class03_11 - 1.017/1.010 Class 11 Multivariate Probability...

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1.017/1.010 Class 11 Multivariate Probability Multiple Random Variables Recall the dart tossing experiment from Class 4. Treat the 2 dart coordinates as two different scalar random variables x and y . In this experiment the experimental outcome is the location where the dart lands. The random variables x and y both depend on this outcome (they are defined over the same sample space). In this case we can define the following events: AB B A y x C y B x A = = = = = I ] ) ( , ) ( [ ] ) ( [ ] ) ( [ ξ ξ ξ ξ y x y x x and y are independent if A and B are independent events for all x and y : P ( C ) = P ( AB ) = P ( A ) P ( B ) Another example … Consider a time series constructing from a sequence of random variables defined at different times (a series of n seismic observations or stream flows x 1 , x 2 , x 3 , …, x n .). Each possible time series can be viewed as an outcome ξ of an underlying experiment. Events can be defined as above: j i j i j j i i ij i i i A A A A x x A x A = = = = I ] ) ( , ) ( [ ] ) ( [ ξ ξ ξ x x x x i and x j are independent if : P ( A ij ) = P ( A i A j ) = P ( A

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class03_11 - 1.017/1.010 Class 11 Multivariate Probability...

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