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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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