Lecture 23. Some topics from 10.1 Fact. Expectation is linear: E( aX ) = a E( X ) for any constant a and any random variable X and E( X + Y ) = E( X ) + E( Y ) for any random variables X and Y . Thus, for any constants a i and random variables X i , E( a 1 X 1 + . . . + a n X n ) = a 1 E( X 1 ) + . . . + a n E( X n ) . Comment. Of course, for the statements in the Fact to be a meaningful, it is necessary that E( X ), E( Y ) and all the E( X i ) exist. In the following, we will deal only with random variables that have and expected value.) Question. Give an example of a discrete random variable that has no expected value. Example. If a die is rolled n times, the expected sum of the rolls is n time the expected outcome of a single roll. More generally, if any experiment with numerical outcomes is repeated n times, the expected sum of the outcomes is n times the expected outcome of a single trial. Example 10.3.
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