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Monte Carlo pp 145-254 | Cite as

Generating Samples

  • George S. Fishman
Part of the Springer Series in Operations Research book series (ORFE)

Abstract

Generating a random sample constitutes an essential feature of every Monte Carlo experiment. If
$$\zeta = \int_R {\varphi (x)} dx,$$
(1)
as in expression (2.67), is to be estimated, then a procedure that randomly generates points from the uniform distribution on ℱm would suffice. When the cost of evaluating φ(x) in Algorithm \({\overline \zeta _n}\) in Sec. 2.7 is prohibitively high, the equivalence of expression(1) and
$$\zeta = \int_{{\mathbb{R}^m}} {\kappa (Z)dF(Z)}$$
(2)
in expression (2.72c) becomes of interest. Recall that, for A⊑ℬ ⊑ℝm, 0≤F(A) ≤FF(ℬ) ≤F(ℝm) = 1. Since F is a distribution function, an alternative procedure randomly samples Z (1),...,Z (n) independently from F and computes
$${\ddot \zeta _n} = {n^{ - 1}}\sum\limits_{i = 1}^n {\kappa ({Z^{(i)}})}$$
as an unbiased estimate of ζ. Although generating Z (i) from F always costs more than generating X (i) from the uniform distribution on ℱm, evaluating k(Z (i)) may be considerably less costly than evaluating φ(X (i)).

Keywords

Generate Sample Pseudorandom Number Generator Reduce Computing Time Logarithmic Evaluation Poisson Generation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • George S. Fishman
    • 1
  1. 1.Department of Operations ResearchUniversity of North CarolinaChapel HillUSA

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