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Unformatted text preview: Example: Let ( X,Y ) and Y be a continuous random vector, with a density f X,Y ( x,y ) = 15 xy 2 , < x,y < 1 , y x. Are X and Y independent? Functions of independent variables If X 1 ,X 2 ,...,X n are independent random vari ables, then the random variables g 1 ( X 1 ) , g 2 ( X 2 ) ,... , g n ( X n ) are also independent. More generally, if we split independent random variables into disjoint groups, then functions of different groups are also independent. The Expected Value Recall that the expected value, or the expec tation, or the mean of a random variable is the weighted average of its values, with the weights being the likelihoods of the values. Notation : E ( X ) or EX . Often also X or simply are used. For a discrete random variable X with a pmf p X the expected value is defined by EX = X a i a i p X ( a i ) . For a continuous random variable X with a pdf f X the expected value is defined by EX = Z  xf X ( x ) dx. Suppose X is a nonnegative random random variable. This means thatvariable....
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 Fall '10
 SAMORODNITSKY

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