Chapter05_answer

Chapter05_answer - Joint distributions Often times, we are...

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Joint distributions Often times, we are interested in more than one  random variable at a time. For example, what is the probability that a car will  have at least one engine problem and at least one  blowout during the same week? X  = # of engine problems in a week Y  = # of blowouts in a week P ( X  ≥ 1,  Y  ≥ 1) is what we are looking for To understand these sorts of probabilities, we need to  develop joint distributions.
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Discrete distributions discrete joint probability mass function  is given  by  f ( x , y ) = P( X  =  x Y  =  y ) where all ( , ) all ( , ) all ( , ) 1.  ( , ) 0  for all  , 2.  ( , ) 1 3.  (( , ) ) ( , ) 4.  ( ( , )) ( , ) ( , ) x y x y A x y f x y x y f x y P X Y A f x y E h X Y h x y f x y = = =
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Return to the car example Consider the following joint pmf for  X  and  Y P ( X  ≥ 1,  Y  ≥ 1) =  P( X  ≥ 1) =    E( X  +  Y ) =  X\Y 0 1 2 3 4 0 1/2 1/16 1/32 1/32 1/32 1 1/16 1/32 1/32 1/32 1/32 2 1/32 1/32 1/32 1/32 1/32
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Joint to marginals The probability mass functions for  X  and  Y   individually (called marginals) are given by Returning to the car example: f X ( x ) =  f Y ( y ) = E ( X ) =  E ( Y ) =  all  all  ( ) ( , ),    ( ) ( , ) X Y y x f x f x y f y f x y = =
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Chapter05_answer - Joint distributions Often times, we are...

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