Probability Theory, 2nd Edition

5 Expected value, variance, and covariance

5.1 Expected value

5.1.1 Expected value of discrete random variables

What is an expected value (also called mean value or expectation) of a random variable? How is it defined? Which property of the random variable does it describe and how it can be computed? Does every random variable possess an expected value? To answer these questions, let us start with an example.

Example 5.1.1.

Suppose N students attend a certain exam. The number of possible points is 100. Given $\mathit{j}=0\mathrm{,}1\mathrm{,}\dots \mathrm{,}100$, let ${\mathit{n}}_{\mathit{j}}$ be the number of students who achieved j points. Now choose randomly, according to the uniform distribution (a single student is chosen with probability $1\mathrm{/}\mathit{N}$), one student. Name him or her ω, and define $\mathit{X}\mathrm{(}\mathit{\omega }\mathrm{)}$ as the ...