Mfs are univariate distributions for both x and y as

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Unformatted text preview: ubset of S The marginal p.m.f.s are univariate distributions for both X and Y as defined by, f (x , y ) , x ∈ S1 (1) f (x , y ) , y ∈ S2 f1 (x ) = (2) y f2 (y ) = x Note that X and Y are independent if f (x , y ) = f1 (x ) f2 (y ) Culpepper, SA STAT 400: Statistics and Probability I 4.1: Distributions of two random variables 4.2: The Correlation Coefficient Discrete Definitions Discrete Examples Continuous Definitions Continuous Examples Discrete Expectations Let u (x , y ) be a function of the random variables X and Y . The expected value of u (x , y ) is, E [u (x , y )] = u (x , y ) f (x , y ) (x ,y )∈S where (x ,y )∈S |u (x , y )| f (x , y ) converges and is finite. If u (x , y ) = x , E [u (x , y )] = E (x ) 2 If u (x , y ) = (x − E (x ))2 , E [u (x , y )] = σX We are also interested in u (x , y ) = xy for understand the linear association between X and Y . Culpepper, SA STAT 400: Statistics and Probability I (3) 4.1: Distributions of two random variables 4.2: The Correlation Coefficient Discrete Definitions Discrete Examples Continuous Definitions Continuous Examples Discrete Examples S...
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