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.

No human investigation can be called real science if it cannot be demonstrated mathematically. Leonardo da Vinci (1452–1519)