Two Random Variables X and Y Are Independently Distributed If $\operatorname { Pr } ( Y = y \mid X = x ) = \operatorname { Pr } ( Y = y )$

Two random variables X and Y are independently distributed if all of the following conditions hold, with the exception of a. $\operatorname { Pr } ( Y = y \mid X = x ) = \operatorname { Pr } ( Y = y )$ .
b. knowing the value of one of the variables provides no information about the other.
c. if the conditional distribution of $Y$ given $X$ equals the marginal distribution of $Y$ .
d. $\quad E ( Y ) = E [ E ( Y \mid X ) ]$ .

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