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Expected Value and Variance for students

Expected Value and Variance for students - Expected Value...

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Expected Value and Variance *Classic Probability asserts that the expected value of a random variable is the long-run average value of the random variable in independent observations. Expected value The expected value of a discrete random variable X, denoted E[X], is defined by E[X]= x X x p x ) ( * In words, the expected value of a discrete random variable is a weighted average of its possible values, and the weight used is its probability. Sometimes, the expected value is referred to as the expectation, the mean, or the first moment. Also, sometimes it is denoted as X . For a function of x g(x), we have the following: E[g(X)]= ( )* ( ) X x g x p x . So, if we define g(x) = 2 x , we have E[ 2 x ] = 2 * ( ) X x x p x . Siblings (x) Probability ) ( x p X X* ) ( x p X 0 .200 1 .425 2 .275 3 .075 4 .025 1.000 As we can see, the expected value of our sibling example is __________. This implies that if we randomly select a student then the expected number of siblings they have is ___________. Clearly a student can’t have ___________ siblings. However, if we continue to select randomly students, the average number of siblings we obtain will be close to ___________.
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Proposition 7.1 Consider a finite population and a variable defined on it. Suppose that a member is selected at random from the population and let X denote the value of the variable for the member obtained. Then the
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