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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