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''.

*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 .

- We consider the square
- 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
- Then

- 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.

- method for the
- 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

- Furthermore, the SLLN (see Theorem WR-5.15)
implies
- that the arithmetic mean
- 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.

- that the arithmetic mean
- 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 .

- Use a random number generator to generate pseudo-random
numbers
that are realizations of random variables being uniformly distributed on .

*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 .

- Hence
is an
- 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 .

- Let
be a continuous function.