Topic_5 - Topic 5 Joint distributions and the CLT Joint...

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1 Topic 5 - Joint distributions and the CLT Joint distributions - pages  145 - 156   Central Limit Theorem - pages  183 - 185  
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2     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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3 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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4 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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5 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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Continuous distributions A joint probability density function for two continuous random  variables, ( X , Y ), has the following four properties: = = = - - - - 1.  ( , )
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This note was uploaded on 03/19/2012 for the course MATH MATH 304 taught by Professor Young during the Spring '09 term at Texas A&M.

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Topic_5 - Topic 5 Joint distributions and the CLT Joint...

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