Fundamentals of Applied Probability and Random Processes, 2nd Edition

Chapter 3

Moments of Random Variables

Abstract

This chapter discusses moments of random variables including the concepts of expectation and variance, higher moments, conditional expectation, and the Chebyshev and Markov inequalities.

Keywords

Mean

expected value

variance

standard deviation

Markov inequality

Chebyshev inequality

3.1 Introduction

Given the set of data X₁, X₂, …, X_N, we know that the arithmetic average (or arithmetic mean) is given by

$\overline{\mathit{X}}=\frac{{\mathit{X}}_1+{\mathit{X}}_2+\dots +{\mathit{X}}_{\mathit{N}}}{\mathit{N}}$

When the above numbers occur with different frequencies, we usually assign weights w₁, w₂, …, w_N to them and the so-called weighted arithmetic mean becomes

$\overline{\mathit{X}}=\frac{{\mathit{w}}_1{\mathit{X}}_1+{\mathit{w}}_2{\mathit{X}}_2+\dots +{\mathit{w}}_{\mathit{N}}{\mathit{X}}_{\mathit{N}}}{{\mathit{w}}_{}}$