Applications of Mathematical Expectation
The idea of an average is especially pertinent to the subject of random variables and readily lends itself to broad development. By the ordinary rule, the arithmetic average of a set of N numbers x 1, x 2, x N is obtained by computing their sum and then dividing by N; that is, \(\bar x\) = (x 1 + x 2 + ··· + x N )/N. Now since it is not necessary that these numbers all be different, let us suppose, in general, that there are n distinct values, x 1, x 2, •••, x n respectively occurring N 1, N 2, •••, N n times, where N 1 + N 2 + ••• + N n = N. Then the sum of the N numbers could be found by adding up the products N 1 x 1 N 2 x 2, •••, N n x n and the arithmetic average would be obtained by dividing the result by N.
KeywordsProbability Density Function Mathematical Expectation Expected Profit Profit Function Unimodal Distribution
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