9.1 Introduction

The term Monte Carlo method or Monte Carlo technique stirs excitement in every engineer and scientist. In this chapter we present rudimentary principles involved in the Monte Carlo method or technique. The Monte Carlo technique is a branch of experimental mathematics, and it is extending into wider applications in such fields as nuclear physics, molecular chemistry, population demographic study, hydrographic analysis, corporate business planning, product quality control, and political opinion survey.

During the infancy period of Monte Carlo development, engineers were introduced to the "hit-or-miss" method analogy, which was greeted with a lukewarm reception. As the development progressed, many researchers in many diversified fields have recognized the great utility of this new experimental mathematics. Hundreds of research papers have been published in science, engineering, economics, and political science.

This chapter covers the very minimum basics of the Monte Carlo method through a few examples for the benefit of beginners without rigorous mathematical proof.

The Monte Carlo method may be classified by a set of particular techniques adopted to solve a problem on hand:

1. Hit-or-miss method;

2. Ordered sample method;

3. Sample mean method;

4. Importance sampling method;

5. Correlated sampling method;

6. Control variate method;

7. Stratified sampling method;

8. Antithetic variate method;

9. Some others.

We shall limit our study to the first four. For other advanced methods, see [1-11]. We summarize the mathematic principle involved and the limitation of each method through examples.

9.2 Hit-or-Miss Method

Suppose we draw an irregular curve on a unit square as shown in Figure 9.1, and wonder how...