Probability Theory and Applications pp 85-123 | Cite as

# Applications of Mathematical Expectation

## Abstract

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*.

## Keywords

Probability Density Function Mathematical Expectation Expected Profit Profit Function Unimodal Distribution## Preview

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