- Write a program that implements the ``hit and miss'' Monte
Carlo integration
algorithm. Find the estimate for the integral of

as a function of , in the interval . Choose , and sample only the -dependent part , and multiply the result by 4. Calculate the difference between and the exact result . This difference is a measure of the error associated with the Monte Carlo estimate. Make a log-log plot of the error as a function of . What is the approximate functional deppendece of the error on for large ? - Estimate the integral of using the simple Monte Carlo
integration by averaging over points, using
(270), and compute the error as a function of , for
upt to 10,000. Determine the approximate functional deppendence of the
error on for large . How many trials are necessary to determine
to two decimal places?
- Perform 10 measurements , with using different
random sequences. Show in a table the values of and
according to (270) and (271). Use
(272) to estimate the standard deviation of the means, and
compare to the values obtained from (273) using the 100,000
values.
- To verify that your result for the error is independent of the
number of sets you used to divide your data, repeat the previous item
grouping your results in 20 groups of 5,000 points each.