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### Simple Applications; Monte-Carlo Estimators

First we recall two simple problems that can be solved by means of Monte-Carlo simulation and have already been discussed in the course ,,Wahrscheinlichkeitsrechnung''.

1. Algorithm to determine the number • A simple computer algorithm for the Monte-Carlo simulation of is the following improved version of Buffon's needle experiment; see Sections 2.5 and 5.2.3 of the course ,,Wahrscheinlichkeitsrechnung''.
• This algorithm is based on the following geometrical facts.

• We consider the square • the circle inscribed into , where • and arbitrarily toss a point into the set .
• Translated into the language of stochastics this means:
• We consider two independent random variables and that are uniformly distributed on the interval and
• determine the probability of the event i.e. that the ,,random point'' is in .

• Then where and denote the area of and , respectively.
• Similarly to Buffon's needle experiment the equation yields a
• method for the statistical estimation of ,
• which is based on the strong law of large numbers (SLLN) and can be easily implemented.
• Let be independent and identically distributed random vectors,
• whose distribution coincides with the one of • and which are regarded as a stochastic model for (independent) experiments.
• Then where are independent and identically distributed random variables with expectation .
• Furthermore, the SLLN (see Theorem WR-5.15) implies
• that the arithmetic mean converges to almost surely.
• Thus, is an unbiased and (strongly) consistent estimator for ,
• i.e., the probability of to be a good approximation for is very high if is large.

• For the implementation of this simulation algorithm one can proceed as follows
• Use a random number generator to generate pseudo-random numbers that are realizations of random variables being uniformly distributed on .

• Put and for .
• Define • Compute .

2. Monte Carlo Integration
• Let be a continuous function.
• Our goal is to find an estimator for the value of the integral that can be determined by Monte-Carlo simulation.
• We consider the following stochastic model.
• Let the random variables be independent and uniformly distributed on , with probability density given by • Let for all .
• By the transformation theorem for independent and identically distributed random variables (see Theorem WR-3.18) the random variables are independent and identically distributed
• with • Furthermore the SSLN (see Theorem WR-5.15) implies that for  • Hence is an unbiased and (strongly) consistent estimator for ,
• i.e., the probability for to be a good approximation of the integral is high for sufficiently large .

• For the implementation of this simulation algorithm one can proceed similarly to Example 1:

• Use a random number generator to generate pseudo-random numbers that are realizations of random variables being uniformly distributed in .
• Define for .
• Compute .    Next: Linear Congruential Generators Up: Generation of Pseudo-Random Numbers Previous: Generation of Pseudo-Random Numbers   Contents
Ursa Pantle 2006-07-20