Random Number Generators

Description

• A random number is a number chosen as if by chance from some specified distribution such that selection of a large set of these numbers reproduces the underlying distribution.
• Almost always, such numbers are also required to be independent, so that there are no correlations between successive numbers. The initial number used as the starting point of a random number generator is known as a seed.
• Uniform Deviate is the basis of all other forms of Random Number Generation, i.e. Normal, Exponential, Poisson, Gamma and Binomial.

• Uniform deviates are random numbers that lie in a range (typically 0 to 1), with any one number in the range just as likely as any other, i.e. a distribution which has constant probability is called a uniform distribution.
• Linear Congruence Method is a method for generating random (pseudorandom) numbers using the linear recurrence relation,

• There is good empirical and theoretical evidence that the simple multiplicative congruential algorithm,

• The output of the above algorithm is then shuffled using the Bay’s and Durham shuffling algorithm to remove the lower-order serial correlations [7].

• Transformation methods generalize to more than one dimension. If x₁, x₂,… are random deviates with a joint probability distribution p(x₁, x₂,…)dx₁dx₂…, and if y₁, y₂, … are all the x’s (same number of y’s as x’s), then the joint probability distribution of y’s is,

• Given a normal distribution in a variate X with mean and variance

• Box-Muller
transformation method transforms from a two-dimensional continuous uniform distribution to a 2-dimensional bivariate normal distribution. If x₁ and x₂ are uniformly and independently distributed between 0 and 1, then z₁ and z₂ as defined below have a normal distribution with mean =0 and variance

• The above can be verified by solving for x₁ and x_2,

• Taking the Jacobian determinant yields,

• We first generate the uniform deviate x and then take some prescribed function of it, y(x). The probability distribution of y, denoted by p(y)dy, is determined by the fundamental transformation law of probabilities, which is,

• In the case of exponential deviates, y(x) = -ln(x), and p(x) is as given by,

• The quantity has the probability distribution

• Random generation using Poisson deviates uses the following,

• Random generation using Gamma deviates uses the following,

• Binomial deviate is the most commonly used discrete probability distribution.
• An experiment which follows a binomial distribution will satisfy the following:
• The experiment consists of n identical trials, where n is fixed in advance.
• Each trial has two possible outcomes, S or F, which we denote success and failure
• The trials are independent, so the outcome of one trial has no effect on the outcome of another.
• The probability of success is constant from one trial to another.
• Random generation using Binomial deviates uses the following,

Math<Toolkit> provides a set of classes to generate randoms numbers in various distribution. The classes provided are
           os_num_rand_uniform          uniform random number generator
           os_num_rand_normal           normal random number generator
           os_num_rand_poisson          poisson random number generator
           os_num_rand_gamma          gamma random number generator
           os_num_rand_exponential    exponential random number generator
           os_num_rand_binomial        binomial random number generator

The example code below shows the usage of the above classes to generate random numbers.

#include <ospace/math.h>&


int
main()
  {
  os_math_toolkit math_init;

  int i, NUM_DEV = 5;

  cout << "===== random uniform =====" << endl;
  os_rand_uniform ru( 0, 10 );
  cout << "lower bound = " << ru.lower_bound() << endl;
  cout << "upper bound = " << ru.upper_bound() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << ru.deviate() << " ";
  cout << endl;
  ru.lower_bound( 10 );
  ru.upper_bound( 20 );
  cout << "lower bound = " << ru.lower_bound() << endl;
  cout << "upper bound = " << ru.upper_bound() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << ru.deviate() << " ";
  cout << "\n" << endl;

  cout << "===== random normal =====" << endl;
  os_rand_normal rn( 0, 10 );
  cout << "mean = " << rn.mean() << endl;
  cout << "variance = " << rn.variance() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << rn.deviate() << " ";
  cout << endl;
  rn.mean( 10 );
  rn.variance( 20 );
  cout << "mean = " << rn.mean() << endl;
  cout << "variance = " << rn.variance() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << rn.deviate() << " ";
  cout << "\n" << endl;

  cout << "===== random poisson =====" << endl;
  os_rand_poisson rp( 10 );
  cout << "mean = " << rp.mean() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << rp.deviate() << " ";
  cout << endl;
  rp.mean( 20 );
  cout << "mean = " << rp.mean() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << rp.deviate() << " ";
  cout << "\n" << endl;

  cout << "===== random exponential =====" << endl;
  os_rand_exponential re( 10 );
  cout << "lambda = " << re.lambda() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << re.deviate() << " ";
  cout << endl;
  re.lambda( 20 );
  cout << "lambda = " << re.lambda() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << re.deviate() << " ";
  cout << "\n" << endl;

  cout << "===== random gamma =====" << endl;
  os_rand_gamma rg( 10 );
  cout << "order = " << rg.order() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << rg.deviate() << " ";
  cout << endl;
  rg.order( 20 );
  cout << "order = " << rg.order() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << re.deviate() << " ";
  cout << "\n" << endl;

  cout << "===== random binomial =====" << endl;
  os_rand_binomial rb( 10, 0.33 );
  cout << "trials = " << rb.trials() << endl;
  cout << "probability = " << rb.probability() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << rb.deviate() << " ";
  cout << endl;
  rb.trials( 20 );
  rb.probability( 0.66 );
  cout << "trials = " << rb.trials() << endl;
  cout << "probability = " << rb.probability() << endl;
  cout << "deviates = ";
  for ( i = 0; i < NUM_DEV; i++ )
    cout << rb.deviate() << " ";
  cout << "\n" << endl;


  return 0;
  }

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