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Statistical computer simulations and Monte Carlo methods.

ABSTRACT

In this paper we present Monte Carlo methods for estimating distribution characteristics of random variables. We describe algorithms for computer simulations of some common distributions and their implementation in MATLAB. The paper concludes with examples and applications.

AMS Subject Classification: 60E05, 60G25, 60G99, 60J10, 62P99, 65C05, 65C60.

KEYWORDS: Monte Carlo methods, random variables, random number generators, computer simulations, MATLAB.

1. Introduction

In this day and age, modelling of phenomena has become necessary in every field of research. Scientists in every area, try to describe what they see, to explain what is observed and to use this knowledge to make inferences and predictions about events occurring in the world we live in. Probabilistic and statistical methods are employed in every research area. These are methods by which decisions are made based on the analysis of data gathered through carefully designed experiments. This entails building of models that closely resemble real life situations, studying the models, developing computational algorithms and procedures and, based on those, making inferences about unknown characteristics. Experimental Statistics provides models that allow us to predict the behavior of a complex system, before actually building or trying it out in real life. It also allows us to assess the probability of error.

In real practice, however, one often deals with rather complicated random phenomena, where computation of probabilities and other characteristics of interest is far from being straightforward. When direct computation is too complicated, resource or time consuming, too approximate, or simply not feasible, we make use of Monte Carlo methods. These are based on computer simulations involving random numbers. If we can write a computer code that replicates a certain phenomenon, we can simulate it any number of times (thousands or millions) and draw conclusions about its real life behavior. The longer run is simulated, the more accurate our predictions are. Monte Carlo methods can be used for computation of probabilities, expected values and other distribution characteristics.

2. Random variables; Definitions and classifications

Definition 1.1 The set of all possible outcomes of an experiment is called the sample space of that experiment and is denoted by S. Its elements are called elementary events. An event is a collection of elementary events, i.e. a subset of S.

Definition 1.2 A collection of events K [??] S is called a [sigma]-field (or [sigma]-algebra) on the sample space S, if it satisfies the conditions

(i) K [not equal to] [empty set];

(ii) A [member of] K [??] [bar.A] [member of] K;

(iii) [A.sub.1],[A.sub.2],...,[A.sub.n] [member of] K [??] [A.sub.1] [union] [A.sub.2] [union]...[union] [A.sub.n] [member of] K.

Definition 1.3 Let s be a sample space and K [??] S a [sigma]-field on it. Probability is a function P: K [right arrow] [??] satisfying the conditions

(i) P (S) = 1;

(ii) P (A) [greater than or equal to] 0, [for all] A [member of] K;

(iii) (iii) P ([A.sub.1] [union] [A.sub.2] [union] ... ) = P ([A.sub.1]) + P ([A.sub.1]) + ...,for any finite or countable collection of mutually exclusive (disjoint) events in K;

The triplet (S, K, P) is called a probability space.

Definition 1.4 Let (S, K, P) be a probability space.

A random variable is a function X: S [right arrow] [??] for which the inverse image [X.sup.-1]((-[infinity], x]) = {e [member of] S | X(e) [less than or equal to] x} [member of] k. If the set of values that X takes is at most countable, then X is a discrete random variable, if it is an interval in [??], then X is a continuous random variable

The function F: [??] [right arrow] [??] defined by F(x)= P (X [less than or equal to] x) is called the cumulative distribution function (cdf) of X.

If X is a discrete random variable, then a better way of describing it is to give its probability distribution function (pdf), an array that contains all its values [x.sub.i], and the corresponding probabilities with which each value is taken [p.sub.i] = P(X = [x.sub.i])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

In this case, the cdf F(x)= [[summation].sub.i[member of]I](hence the name).

If X is a continuous random variable, then its cdf is absolutely continuous, which means that there exists a function f : [??] [right arrow] [??] such that F(x) = [[integral].sup.x.sub.-[infinity]]f(t)dt. That function f is called the probability density function (pdf) of the continuous random variable X.

So, either way (the discrete or the continuous case), the pdf is what describes a random variable.

Remark 1.5 Although in both cases we call it generically a pdf, the word distribution emphasizes a discrete behaviour, whereas density suggests a continuous set. However, some authors do not make the distinction between the two cases and call it a probability density function in both cases.

2.1. Common discrete distributions

For describing discrete random variables associated with experiments, an important notion is that of Bernoulli trials. There are three characteristics that define such trials:

- they are independent;

- each trial has two possible outcomes, which we refer to as "success" and "failure", respectively;

- the probability of success, p, is the same in every trial (and then we denote the probability of failure by q = 1 - p ).

Note that the second condition above is not at all restrictive, we simply divide the sample space into "whatever is of interest to us" (and call that success) and everything else (which will constitute failure).

Bernoulli distribution with parameter p [member of] (0, 1), Bern(p). This is the simplest of distributions, with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

It can be used to model a Bernoulli trial.

Binomial distribution with parameters n [member of] [??], p [member of] (0, 1), B(n, p). Consider a series of n Bernoulli trials with probability of success p in every trial. Let X be the number of successes that occur in the n trials. That is a Binomial distribution, with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Remark 1.6 Note that a binomial B(n, p) variable is the sum of n independent Bern(p) variables

Geometric distribution with parameter p [member of] (0, 1), G(p). Consider an infinite sequence of Bernoulli trials with probability of success p in every trial. Let X denote the number of failures that occurred before the first success. That is a Geometric distribution, with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Pascal (Negative Binomial) distribution with parameters n [member of] [??], p [member of] (0, 1), NB(n, p). Just as before, consider an infinite sequence of Bernoulli trials with probability of success p in every trial. Let X denote the number of failures that occurred before the nth success. That is a Negative Binomial distribution, with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

Remark 1.7 Note that a negative binomial NB(n, p) variable is the sum of n independent G(p) variables.

Poisson distribution with parameter [lambda] > 0, [??]([lambda]), with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

Such a variable is defined in the context of a Poisson process: a process in which discrete events are observed in a continuous medium (length, aria, volume or time). Examples would be: messages that arrive within an hour, white blood cells existing in a drop of blood, earthquakes that happen in an area during a year, etc. Such events are called rare events, because they are extremely unlikely to occur simultaneously or within a short interval (of time, length, etc.). The Poisson variable X denotes the number of such rare events that occur in a given interval of the continuous medium. As that number increases (tending to [infinity]), the probability of that many rare events occurring tends to 0 (since they are rare...). The parameter of the Poisson distribution, [lambda], represents the average number of the considered rare events per unit (of time, length, etc.).

2.2. Uniform distribution

The continuous Uniform distribution plays a unique role in statistical modelling, in the sense that any random variable (with any pdf), discrete or continuous, can be generated from a uniformly distributed variable, as we shall see in the next section. The pdf of a Uniform distribution U(a, b), of parameters -[infinity] < a < b < [infinity], is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and is used whenever a value is picked "at random" from an interval, in situations when all values from an interval are equally probable to be taken by a random variable.

By integration, its cdf is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

An important special case is that of a Standard Uniform distribution U(0,1), with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

and cdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

3. Monte Carlo methods

As mentioned before, Monte Carlo methods are used in simulating random phenomena and effectively estimating distribution characteristics. For the distributions discussed in Section 2.1, we generate a large number of simulations and estimate probabilities by long-run relative frequencies.

3.1. Simulation of Bernoulli and Binomial random variables

Bernoulli distribution

As we mentioned in Section 2.2., a Standard Uniform U(0,1) variable can be used to model a binary process, that is, a Bernoulli trial. Let p [member of] (0, 1) be fixed and let U [member of] U(0,1). Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We consider a "success" the event X = 1 and a "failure" the event X = 0. Then, by (2.8), we have

P(success) = P(X = 1) = P(U < p) = [F.sub.U](p) = p

Thus, X [member of] Bern(p) and we have generated a Bernoulli random variable.

Example 3.1 Consider the experiment of rolling a die. The event of interest, i.e. "success", is getting a 6.

Below is a MATLAB (1) code generating N = 10000 simulations of a Bernoulli random variable with parameter p = 1/6.

% Simulate Bernoulli distribution Bern(p).

p = 1/6; % the parameter of the Bern distr.

N = 1e5; % the number of simulations

U = unifrnd(1,N); % sequence of N U(0,1)random numbers

X = (U < p); % simulation of N Bernoulli trials

% Compare it to the Bern(p) distribution

U_X = unique(X) % the values of X listed ONLY ONCE, no % repetitions

n_X = hist(X,length(U_X)); % the frequency of each value in U_X % (how many times each occurs)

rel_freq = n_X/N % the relative freq. n_X/N approximates the % probability

The run of this code will yield the results

U_X = 0 1

rel_freq = 0.8331 0.1669

The probability p = 1/6 with which X takes the value 1 is well approximated by the relative frequency of the occurrence of the value 1, namely, 0.1669.

Now, having generated a Bernoulli variable, we can use that to simulate other discrete distributions described in Section 2.1.

Binomial Distribution

We use Remark 1.6. We generate n independent Bern(p) and add them to obtain a Binomial B(n, p) distribution.

Example 3.2 A lab network consists of 20 computers. Twenty five percent of the computers are in need of updated antivirus software. Let X denote the number of computers in the lab who have the latest antivirus software installed.

Then, "success" here means that a computer is up-to-date in antivirus protection and the probability of success is p = 0.75. The number of trials is n = 20 . The variable X denotes the number of successes occurred in the n trials and, therefore, has a Binomial B(20, 0.75) distribution. The code for the simulation of this problem is given below. This time, instead of looking at each individual value of the variable and its relative frequency, we compare it graphically to the actual Binomial distribution considered (see Figure 1).

% Simulate Binomial distribution B(n,p).

n = 20;

p = 0.75 % the parameters of the Binomial distr.

N = 1e5; % the number of simulations for i = 1 : N

U = unifrnd(1,n); % a sequence of n U(0,1)random numbers

X(i) = sum(U < p); % generate a B(n,p) variable as the sum % of the n Bernoulli variables

end

% Compare it to the B(n,p) distribution.

k = 0 : n; % all the values of a Binomial distr.

p_k = binopdf(k,n,p); % the probabilities of a Binomial distr.

U_X = unique(X); % the values of X listed ONLY ONCE, no % repetitions

n_X = hist(X,length(U_X)); % the frequency of each value in U_X (how many times each occurs)

rel_freq = n_X/N; % the relative freq. n_X/N approximates the % probability

3.2. Simulation of Geometric and Negative Binomial random variables Geometric distribution

A Geometric random variable represents the number of failures occurred before the first success, in an infinite sequence of Bernoulli trials. Therefore, we simulate Bernoulli trials and count the number of failures before the occurrence of the first success.

Example 3.3 A computer virus can damage a file in the system it has entered with probability 0.6. The system manager checks the files for this virus in order to clean them. Let X denote the number of files that were found to be clean before the first infected file is found.

Define "success" as a file being infected by this virus. Then variable X has a Geometric distribution with parameter p = 0.6. The code for simulating this variable and the graphical comparison with the G(0.6) distribution (Figure 2) are given below.

% Simulate Geometric distribution G(p).

p = 0.6 % parameter of the Geometric distr.

N = 1e5; % the number of simulations for i = 1 : N

X(i) = 0; % initial number of failures

U = unifrnd(1,1);

while U >= p % "U < p" represents success, so "U >= p" % represents failure

X(i) = X(i) + 1; % add the number of failures

end % stop at the first success

end

% Compare it to the G(p) distribution

k = 0 : 20; % the relevant values of a Geometric distr. (those % with probability > 0); otherwise, there is an infinite number % of values

p_k = geopdf(k,p); % the probabilities of a Geometric distr.

U_X = unique(X); % the values of X listed ONLY ONCE, no % repetitions

n_X = hist(X,length(U_X)); % the frequency of each value in UX

rel_freq = n_X/N; % the relative freq. n_X/N approximates the % probability

Negative Binomial distribution

We make use of Remark 1.7. We generate n independent G(p) variables and add them up.

Example 3.4 Consider, again, the experiment of rolling a die. Let X denote the number of even numbers rolled before the third odd number occurs.

Here, "success" means that an odd number shows on the die and its probability is p = 0.5. Then X has a Negative Binomial distribution with parameters n = 3 and p = 0.5. The code for simulating this variable and the graphical comparison with the NB(3, 0.5) distribution (Figure 3) are given below.

% Simulate Negative Binomial distribution NB(n,p).

n = 0.5;

p = 0.6 % the parameters of the Negative Binomial distr

N = 1e5; % the number of simulations for i = 1 : N

for j = 1 : n

Y(j) = 0; % initial number of failures

U = unifrnd(1,1);

while U >= p % "U < p" represents success, so "U >= p" % represents failure

Y(j) = Y(j) + 1; % add the number of failures

end % stop at the first success

end

X(i) = sum(Y); % generate a NB(n,p) variable as the sum % of the n Geometric variables

end

% Compare it to the NB(n,p) distribution

k = 0 : 25; % the relevant values of a Negative Binomial distr. % (those with probability > 0); otherwise, there is an infinite % number of values

p_k = nbinpdf(k,n,p); % the probabilities of a Geometric distr.

U_X = unique(X); % the values of X listed ONLY ONCE, no % repetitions

n_X = hist(X,length(U_X)); % the frequency of each value in UX rel_freq = n_X/N; % the relative freq. n_X/N approximates the % probability

3.3. Simulation of arbitrary discrete distributions

There are several ways of generating a random variable with an arbitrary discrete pdf. We present here one of the simplest methods, which is based on our earlier simulation of a Bernoulli trial. Let us revisit how a Bernoulli variable was generated. We used a random number U [epsilon] U(0,1) to divide the interval [0, 1] into two parts of lengths p and 1 - p, respectively. Then the value of the Bernoulli variable was determined according to which part of the interval U fell in. Now we will do the same when the interval [0, 1] is divided into an arbitrary number of subintervals. The procedure is described in the following algorithm.

Let X be any discrete random variable with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Algorithm 3.5 Simulation of discrete random variables

1. Divide the interval [0, 1] into the subintervals [{[A.sub.i]}.sub.i[member of]I] as follows

[A.sub.0] = [0, [p.sub.0])

[A.sub.1] = [[p.sub.0], [p.sub.0] + [p.sub.1])

[A.sub.2] = [[p.sub.0] + [p.sub.1], [p.sub.0] + [p.sub.1] + [p.sub.2])[??]

2. Let U [epsilon] U(0,1) be a Standard Uniform random variable.

3. When U [epsilon] [A.sub.i], let X = [x.sub.i].

Indeed, it follows that for each i [epsilon] I, subinterval [A.sub.i] has length [p.sub.i] and, by (2.8), P([p.sub.0] + [p.sub.0] + [??] + [p.sub.i-1] [less than or equal to] U < [p.sub.0] + [p.sub.1] [??] + [p.sub.i]) = [F.sub.U, i-1] = [p.sub.i], where we used the notation [F.sub.U,j] = F([p.sub.0] + [p.sub.1] + [??] + [p.sub.j]).

Remark 3.6

1. Note that Algorithm 3.5 works both for a finite set of values I or for a countably infinite one.

2. The algorithm above can also be used to generate the common distributions discussed in Sections 3.1 and 3.2, but we preferred to use their special properties and the relationship to Bernoulli trials.

Let us use Algorithm 3.5 to generate a Poisson p([lambda]) variable with pdf given by (2.6). The initial value of the its cdf will be [F.sub.U, 0] = [p.sub.0] = [e.sup.-[lambda]] and subsequent values of the cdf are obtained by the recurrence formula

[F.sub.U, i] = [F.sub.U, i-1] + [p.sub.i] = [F.sub.U, i-1] + [[[lambda].sup.i]/i!][e.sup.-[lambda].

Example 3.6 In a certain area, at a local firm, network breakdowns occur every 10 months, on the average. Let X denote the number of such network breakdowns that occur in the next 10 months.

Network breakdowns qualify as rare events, because no two such breakdowns can occur simultaneously. Thus, the number X of breakdowns in the next 10 months has a Poisson distribution with parameter [lambda] = 10. We use Algorithm 3.5 to simulate it. The results of N = 10000 simulations are given below.

% Simulate Poisson distribution P(lambda).

lambda = 10; % the parameter of the Poisson distr

N = 1e5; % the number of simulations

for j = 1 : N

U = unifrnd(1,1); % generated U(0,1) variable

i = 0; % initial value

F(j) = exp(-lambda); % initial value of the cdf F(0)

while (U >= F(j)) % check that U is in A_i, i.e.

% F(i-1) <= U < F(i);

% the while loop ends when U < F(i)

F(j) = F(j) + exp(-lambda) * lambda^i/gamma(i+1); % new value % of F

i = i + 1; % count the values in A_i end

X(j) = i; % one Poisson variable end

Y = sum(X); % N Poisson variables

% Compare it to the P(lambda) distribution.

k = 0 : 27; % the relevant values of a Poisson distr.(those with % probability > 0); otherwise, there is an infinite number of % values

p_k = poisspdf(k,lambda); % the probabilities of a Poisson % distr.

U_X = unique(X); % the values of X listed ONLY ONCE, no % repetitions

n_X = hist(X,length(U_X)); % the frequency of each value in UX rel_freq = n_X/N; % the relative freq. n_X/N approximates the % probability

4. Conclusions

We have described procedures and algorithms for computer simulations of some common discrete random variables. With the use of a Standard Uniform random variable, any distribution can be generated. Then we used these simulations to make inferences about each distribution, namely to estimate probabilities, i.e., we used Monte Carlo methods to approximate probabilities. As all the examples show, with a relatively moderate number of simulations, N = 10000 (moderate, in the sense that for some processes millions such simulations may be required), the results are very good, the approximations quite accurate.

References

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[3] M. Baron, Probability and Statistics for Computer Scientists, 2nd Edition, CRC Press, Taylor & Francis Group, Boca Raton, FL, USA, 2014.

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[5] W. Feller, An introduction to probability theory and its applications, Vol. I-II, John Wiley, New York, 1957, 1966.

[6] H. Lisei, S. Micula, A. Soos, Probability Theory trough Problems and Applications, Cluj University Press, 2006.

[7] S. Micula, Probability and Statistics for Computational Sciences, Cluj University Press, 2009.

[8] J.S. Milton, J. C. Arnold, Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences, 3rd Edition. McGraw-Hill, New York, 1995.

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[10] http://www.mathworks.com/help/matlab/.

Sanda Micula (1*)

(1*) Corresponding author. Department of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, smicula@math.ubbcluj.ro

(1) MATLAB[R] is a trademark of MathWorks Inc.
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Publication:Journal of Information Systems & Operations Management
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