NotesSep17 - Example Let X be a Pareto random variable with...

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Example: Let X be a Pareto random variable with a cdf F X ( x ) = 1 - x - p , x 1 for p > 1, a parameter of the distribution. Find the expectation of X .
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The formula for the expectation we obtained EX = Z 0 (1 - F X ( x )) dx = Z 0 P ( X > x ) dx is valid for any type of nonnegative random variables. For a discrete random variable X with possible values 0 , 1 , 2 , . . . this expression reduces to a more convenient one EX = X n =0 (1 - F X ( n )) = X n =0 P ( X > n ) .
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Example: Let X be a geometric random vari- able with probability for success p . Find the expectation of X .
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Expectation of a function of a random variable Given a X discrete random variable and Y = g ( X ), one can compute the pmf p Y of Y and derive its expected value of Y via EY = X y i y i p Y ( y i ) , but it is unnecessary. One can use the pmf of X instead, and still compute the expectation of Y by EY = X x i g ( x i ) p X ( x i ) .
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If X is a continuous random variable and Y = g ( X ), then EY = Z -∞ g ( x ) f X ( x ) dx.
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