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stat exam 2

# stat exam 2 - all all all 1 0 for a ll 2 1 3 4 x y x y A x...

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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)= Σ (x)[p(x)] E(x+y)= Σ (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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stat exam 2 - all all all 1 0 for a ll 2 1 3 4 x y x y A x...

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