NotesSep15 - Example Let(X Y and Y be a continuous random...

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Example: Let ( X, Y ) and Y be a continuous random vector, with a density f X,Y ( x, y ) = 15 xy 2 , 0 < x, y < 1 , y x . Are X and Y independent?
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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.
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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 μ X or simply μ are used.
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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.
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Suppose X is a nonnegative random random variable. This means that P ( X < 0) = 0.
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