A Primer for the Monte Carlo Method

Ilya M. Sobol

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9780849386732
• eBook ISBN: 9781351469579
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

The Monte Carlo method is a numerical method of solving mathematical problems through random sampling. As a universal numerical technique, the method became possible only with the advent of computers, and its application continues to expand with each new computer generation. A Primer for the Monte Carlo Method demonstrates how practical problems in science, industry, and trade can be solved using this method. The book features the main schemes of the Monte Carlo method and presents various examples of its application, including queueing, quality and reliability estimations, neutron transport, astrophysics, and numerical analysis. The only prerequisite to using the book is an understanding of elementary calculus.

1

simulating random variables

random variables

We assume that the reader is acquainted with the concept of probability, and we turn directly to the concept of a random variable.

The words “random variable,” in ordinary lay usage, connote that one does not know what value a particular variable will assume. However, for mathematicians the term “random variable” has a precise meaning: though we do not know this variable’s value in any given case, we do know the values it can assume and the probabilities of these values. The result of a single trial associated with this random variable cannot be precisely predicted from these data, but we can predict very reliably the result of a great number of trials. The more trials there are (the larger the sample), the more accurate our prediction.

Thus, to define a random variable, we must indicate the values it can assume and the probabilities of these values.

Discrete Random Variables

A random variable ξ is called discrete if it can assume any of a set of discrete values x₁, x₂, …, x_n. A discrete random variable is therefore defined by a table

$\xi \sim \left(\begin{array}{llll}{x}_1\hfill & x_2\hfill & \cdots \hfill & x_n\hfill \\ p_1\hfill & p_2\hfill & \cdots \hfill & p_n\hfill \end{array}\right)$

(T)

where x₁, x₂, …, x_n are the possible values of ξ, and p₁, p₂, …, p_n are the corresponding probabilities. To be precise, the probability that the random variable ξ will be equal to x_i (denoted by P{ξ = x_i) is equal to p_i:

$P\left\{\xi =x_i\right\}=p_i$

Table (T) is called the distribution of the random variable ξ.

The values x₁, x₂, …, x_n can be arbitrary.^* However, the probabilities p₁, p₁, …, p_n must satisfy two conditions:

1. All p_i are positive:

$p_i>0$

(1.1)

2. The sum of all the p_i equals 1:

$p_1+p_2+\cdots +p_n=1$

(1.2)

The latter condition requires that in each trial, ξ must necessarily assume one of the listed values.

The number

$\mathbf{\text{M}}\xi ={\displaystyle \sum _{i=1}^n{x}_i{p}_i}$

(1.3)

is called the mathematical expectation, or the expected value, of the random variable ξ.

To elucidate the physical meaning of this value, we rewrite it in the following form:

$\mathbf{\text{M}}\xi =\frac{{\displaystyle \sum _{i=1}^n{x}_i{p}_i}}{{\displaystyle \sum _{i-1}^n{p}_i}}$

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Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• 1 SIMULATING RANDOM VARIABLES
• 2 EXAMPLES OF THE APPLICATION OF THE MONTE CARLO METHOD
• 3 ADDITIONAL INFORMATION
• REFERENCES
• INDEX