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Unformatted text preview: all ( , ) all ( , ) all ( , ) 1 . ( , ) 0 for a ll , 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 = = = ---- 1. ( , ) 0 for a ll , 2. ( , ) 1 3. (( , ) ) ( , ) 4. ( ( , )) ( , ) ( , ) A f x y x y f x y d xd y P X Y A f x y d xd y E h X Y h x y f x y d xd y = = = ( ) ( , ) , ( ) ( , ) X Y f x f x y dy f y f x y d x -- = = JOINT DISTRIBUTIONS Joint probability distribution: all joint outcomes have at least a 0% chance of occurring and the summation of all joint outcomes equals to 1. E(X) and E(Y) are the weighted averages of the marginals for X and Y respectively. Discrete distributions o A discrete joint probability mass function is given by f ( x , y ) = P( X = x , Y = y ) where E(x)= (x)[p(x)] E(x+y)= (x+y)[p(x+y)] Joint to Marginals X\Y 1 1/2 1/16 .56 1 1/16 1/32 .093 0.56 .093 f x (0)=0.56 E(x)=(0*.56)+(1*.093) Continuous distributions o A joint probability density function for two continuous random variables, ( X , Y ), has the following four properties: o o The marginal pdfs of X and Y can be found by o note: f x with dy and the domain of y. Independence of X and Y o The random variables X and Y are independent if f ( x , y ) = f X ( x ) f Y ( y ) for all pairs ( x , y ). SAMPLING DISTRIBUTIONS A statistic is a function of these random variables that is used to estimate some characteristic of the population distribution....
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This note was uploaded on 10/04/2010 for the course STAT STAT 211 taught by Professor Idontremember during the Spring '10 term at Texas A&M.
- Spring '10