Question

A Monte Carlo simulation is a method for finding a value that is difficult to compute by performing many random experiments.

For example, suppose we wanted to estimate π to within a certain accuracy. We could do so by randomly (and independently) sampling n points from the unit square and counting how many of them are inside the unit circle (assuming that the probability of selecting a point in a given region is proportional to the area of the region). By assuming we actually get the expected number, we can solve for π.

(a) Describe a reasonable sample space to model this experiment.

(b) Let N be the number of sample points that are inside the unit circle. Find E(N).

(c) Use this to construct a random variable P with E(P) = π. This random variable will give your estimate of π.

(d) Find the variance of P.

(e) Use Chebychev’s inequality to find a value of n that guarantees your estimate is within 1/1000 of π with probability at least 50%.

Answer 1

Ans) a) A reasonable sample space to model this experiment is the unit square i.e, -

b) Let N be the number of sample points that are inside the unit circle. Given that the probability of selecting a point in a given region is proportional to the area of the region. Therefore probability of selecting a point in the unit circle is    as area of the unit circle is    and the area of the unit square is 1.

Let there are n sample points.Therefore if a sample point falls inside the unit circle then we have a success, otherwise we have a failure. So,  

c)      

d)   

e) By Chebyshev's inequality

Given   

Let    as we put the estimate of as P .

Alternatively,

Given    So,  

Now    here we are assuming the distribution of P to be normal as n is large

As   

where   is the upper th point of N(0,1)

Here is unknown so we estimate it by

  

otherwise