Monte Carlo Simulation

Abstract

This chapter discusses the basic concept and techniques for Monte Carlo simulation. The simulation methods for a single random variable as well as those for a random vector (consisting of multiple variables) are discussed, followed by the simulation of some special stochastic processes, including Poisson process, renewal process, Gamma process and Markov process. Some advanced simulation techniques, such as the importance sampling, Latin hypercube sampling, and subset simulation, are also addressed in this chapter.

Problems

3.1

Disk Line Picking Problem. Randomly pick up two points on a unit disk (with radius 1), and let D be the distance between the two points. What is the mean value of D?

3.2

A random variable X follows a Rayleigh distribution with scale parameter $\sigma$. Using the inverse transformation method, develop an algorithm to generate a sample for X, x.

3.3

For a Weibull random variable X with a scale parameter of u and a shape parameter of $\alpha$, develop an algorithm to generate a sample for X using the inverse transformation method.

3.4

For a Cauchy random variable X whose PDF is as in Eq. (2.380), develop an algorithm to generate a sample for X using the inverse transformation method.

3.5

Given a positive $\lambda$, we simulate a sequence of independent uniform (0,1) samples $u_1,u_2,\ldots$, and find such an integer n that

$$\begin{aligned} n=\min \left( k: \prod _{i=1}^{k}u_i<\exp (-\lambda )\right) -1 \end{aligned}$$

(3.151)

Argue that n is a sample of Poisson random variable with mean $\lambda$ [17].

3.6

The PDF of random variable X is as follows,

$$\begin{aligned} f_X(x)=\frac{1}{\sqrt{2}\pi ^{\frac{3}{2}}}\int _{-\infty }^{\infty }\exp \left[ -\frac{(x-y)^2}{2}\right] \cdot \frac{1}{y^2+1}\mathrm {d}y,\quad -\infty<x<\infty \end{aligned}$$

(3.152)

Establish a procedure to sample X using the composition method.

3.7

The PDF of random variable X is as follows with $\theta \ge 0$,

$$\begin{aligned} f_X(x)=\frac{x}{\theta +2}\exp \left( -\frac{x^2}{2(\theta +1)}\right) +(\theta +2)x\exp \left( -(\theta +2)x\right) ,\quad x\ge 0 \end{aligned}$$

(3.153)

Establish a procedure to sample X using the composition method.

3.8

If the PDF of random variable X is

$$\begin{aligned} f_X(x)=\frac{4}{41}(x^3+3x^2+6x),\quad x\in [0,1] \end{aligned}$$

(3.154)

establish a procedure to generate a realization of X [Hint. Note that $1+x+\frac{x^2}{2}+\frac{x^3}{6}<\mathrm {e}^x$ when $x\ge 0$.].

3.9

For a Laplace variable X whose PDF is as in Eq. (2.25), establish a procedure to generate a realization of X using the acceptance-rejection method.

3.10

Consider two continuous random variables X and Y with marginal CDFs of $F_X(x)$ and $F_Y(y)$ respectively, and a joint CDF of $C(F_X(x),F_Y(y))$, where C is the Farlie-Gumbel-Morgenstern copula, $C(u,v)=uv+\theta uv(1-u)(1-v)$ with $\theta \in [-1,1]$. Develop an algorithm to simulate a sample pair for (X, Y), (x, y).

3.11

For two correlated random variables $X_1,X_2$ with an identical COV of 0.5, let $\rho _X$ be their linear correlation coefficient. When using the Nataf transformation method to generate a sample pair of $X_1,X_2$, let $Y_1,Y_2$ be the corresponding standard normal variables (that is, $\Phi (Y_i)=F_{X_i}(X_i)$ for $i=1,2$, where $F_{X_i}$ is the CDF of $X_i$), and $\rho _Y$ the linear correlation coefficient for $Y_1,Y_2$. Plot the relationship between $\rho _X$ and $\rho _Y$ for the following two cases respectively.

(1) Both $X_1$ and $X_2$ follow an Extreme Type I distribution.

(2) $X_1$ is a normal variable while $X_2$ follows an Extreme Type I distribution.

3.12

Consider three identically lognormally distributed variables $X_1, X_2$ and $X_3$ with a mean value of 2 and a standard deviation of 0.5. Let $\rho _{ij}$ be the correlation coefficient of $X_i,X_j$ for $i,j\in \{1,2,3\}$. When $\rho _{ij}=\exp (-|i-j|)$, how to sample a realization for $X_1,X_2,X_3$ based on the Nataf transformation method? State the detailed steps.

3.13

For two correlated Poisson variables $X_1$ and $X_2$ with mean values of 1 and 2 respectively, let $\rho _{12}$ be their correlation coefficient. When it is applicable to use Eq. (3.62) to generate a sample pair for $X_1,X_2$, what is the range for $\rho _{12}$?

3.14

With a reference period of [0, T], the non-stationary Poisson process is used to model the occurrence of a sequence of repeated events. For the following two cases, establish a method to simulate a sequence of occurrence times of the events.

(1) The occurrence rate is $\lambda (t)=2+\sin (at)$, where $a>0$ is a time-invariant parameter.

(2) The occurrence rate is $\lambda (t)=\exp (bt)$, where $b>0$ is a time-invariant parameter.

3.15

With a reference period of [0, T], the non-stationary Poisson process is used to model the occurrence of a sequence of repeated events. If the occurrence rate is

$$\begin{aligned} \lambda (t)=\left\{ \begin{aligned}&2+\sin (at),&t\in [0, T_1] \\&\exp (bt),&t\in [T_1,T] \end{aligned}\right. \end{aligned}$$

(3.155)

where $0<T_1<T$, a and b are two positive constants, establish a method to simulate a sequence of occurrence times of the events.

3.16

Recall Example 2.47. For a reference period of 50 years (i.e., $T=50$ years), assume that the damage loss conditional on the occurrence of one cyclone event has a (normalized) mean value of 1 and a COV of 0.4. The cyclone occurrence is modelled as a stationary Poisson process with a mean occurrence rate of 0.5/year. Use a simulation-based approach to verify the mean value and variance of $D_c$ as obtained in Example 2.47.

3.17

In Problem 3.16, if the cyclone occurrence is modelled as a non-stationary Poisson process with a mean occurrence rate of $\lambda (t)=0.5(1+0.01t)$, where t is in years, what are the simulation-based mean value and variance of $D_c$? Compare the results with those in Problem 3.16.

3.18

Let N be a non-negative integer random variable with a mean value of $\mu _N$, and $X_1,X_2,\ldots X_N$ a sequence of statistically independent and identically distributed random variables (independent of N) with a mean value of $\mu _X$. Show that

$$\begin{aligned} \mathbb {E}\left( \sum _{i=1}^{N} X_i\right) =\mu _N\mu _X \end{aligned}$$

(3.156)

Equation (3.156) is known as the Wald’s equation.

3.19

The cables used in a suspension bridge have a lognormally distributed service life with a mean value of 40 years and a COV of 0.15.

(1) In order to guarantee the serviceability of the bridge in a long service term, at which rate does each cable have to be replaced by a new one?

(2) Use a simulation-based method to verify the result in (1).

3.20

In Problem 3.19, we additionally consider the impact of earthquakes on the cable serviceability. Suppose that the occurrence rate of earthquakes is a stationary Poisson process with a rate of 0.02/year, and that the cable is immediately replaced after each earthquake.

(1) In order to guarantee the serviceability of the bridge in a long service term, at which rate does each cable have to be replaced by a new one?

(2) Use a simulation-based method to verify the result in (1).

3.21

A structure can be categorized into one of the following four states: state 1 no damage; state 2 minor damage; state 3 moderate damage; state 4 severe/total damage. Let $X_1,X_2,\ldots$ be a stochastic process that represents the yearly state of the structure. Modeling the sequence $X_1,X_2,\ldots$ as a Markov chain, the transition matrix is as follows, where $P_{ij}$ means the probability of the structure being classified as state j provided that the previous state is i for $i,j\in \{1,2,3,4\}$.

$$\begin{aligned} \mathbf {P}=[P_{ij}]=\begin{bmatrix} 0.90 &{} 0.05 &{} 0.05 &{} 0 \\ 0 &{} 0.90 &{} 0.06 &{} 0.04 \\ 0 &{} 0 &{} 0.80 &{} 0.20 \\ 0 &{} 0 &{} 0 &{} 1 \end{bmatrix} \end{aligned}$$

(3.157)

(1) If the structure is in state 1 at initial time, what is the probability of state 3 after five (5) years?

(2) If the structure is in state 2 at initial time, what is the probability of state 4 after ten (10) years?

(3) Show that after many years, the structure will be in state 4 with probability 1.

3.22

Consider the serviceability of a lining structure subjected to water seepage, as discussed in Problem 2.32. Assume that the hydraulic conductivity K and the water pressure p are two independent random variables. The structural performance is deemed as “satisfactory” if the water seepage depth at time t, $\chi (t)$, does not exceed a predefined threshold $\chi _{\mathrm {cr}}$. Establish a procedure to compute the probability of satisfactory structural performance at time $t_0$.

3.23

In Problem 3.22, if $\chi _{\mathrm {cr}}=0.15$ m, the hydraulic conductivity K is lognormally distributed having a mean value of $5.67\times 10^{-13}$ m/s and a COV of 0.3, and the water pressure is a Gamma variable with a mean value of 0.06 MPa and a COV of 0.2, what is the probability of satisfactory structural performance at the end of 50 years? Note that $\omega _0$ is the water density (997 kg/m³).

3.24

Reconsider the portal frame in Example 2.7. Develop a simulation procedure to estimate the probability that V occurs only conditional on the failure mode of “beam”.

3.25

Estimate the following integral using Monte Carlo simulation method,

$$\begin{aligned} f=\int _{0}^{\infty }\int _{0}^{z}\int _{0}^{y} xyz^2\exp \left( -\frac{x^2}{2}-y-z\right) \mathrm {d}x\mathrm {d}y\mathrm {d}z \end{aligned}$$

(3.158)

3.26

In Problem 3.25, numerically show that the use of importance sampling will improve the calculation efficiency.

3.27

Suppose that a normal random variable X has a mean value of 2 and a standard deviation of 1. Using a simulation-based approach to estimate $\mathbb {E}(X^2)$, show that the Latin hypercube sampling performs better than the brute-force Monte Carlo simulation in terms of convergence rate.

3.28

Suppose that X is a Gamma variable with a mean of 3 and a standard deviation of 1; Y is a lognormal variable with a mean of 2 and a standard deviation of 0.7. Use the subset simulation method to estimate the probability $\mathbb {P}[\exp (-X)\cdot Y>1.5]$.