Estimating e using Monte Carlo

[Candyfloss]

Hey everyone. I am new to c++. I am really confused as to how to get the estimation of the number e using Monte Carlo. I have searched on Monte Carlo but I still don't understand. Please explain the method of Monte Carlo and help with how to use that to estimate e. The assignment also said that the output must be dynamically calculated. What does that mean and how do you do so?

Thanks

[dhayden]

This is more a question to ask google than a C++ forum. The very first article when I google "estimate e monte carlo" is:
https://stats.stackexchange.com/questions/193990/approximate-e-using-monte-carlo-simulation

Read up on that. Once you have some C++ code that is giving you trouble, you should post here.

[lastchance]

The Monte Carlo method is to use the mean of a large number of experiments with known probability distribution to estimate some quantity. It is based on the assumption that if you do a lot of trials then your average value should be close to the mean or expectation of that probability distribution.

@dhayden's link is excellent, but I've just spent over an hour failing to prove the relevant theorem, so I'll just have to accept the end result.

Your "experiment" here is conceptually simple: just add up successive values of a variable uniformly distributed on [0,1] and return the number of them when their sum first exceeds 1.0. The theoretical average of the probability distribution is (apparently) e, so your average over a large number of experiments should approach this.

My suggestion:
- have a double myRand() function to return a single value uniformly distributed on [0,1]. You can borrow some ideas from
http://www.cplusplus.com/reference/random/uniform_real_distribution/

- have an int experiment() function to carry out the simulated experiment and return the number of variables when their sum first exceeds 1.

- int main() will then average the result of however many experiments you want.


Although I couldn't prove the result (I'm gutted!) it certainly works:

    Trials   Estimate
         1          2
        10        3.2
       100       2.54
      1000      2.723
     10000     2.7465
    100000    2.71662
   1000000   2.718129
  10000000  2.7188502

Actual value of e is 2.7182818

More web links at
https://en.wikipedia.org/wiki/E_(mathematical_constant) - see: stochastic representations, near the bottom
http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LawOfLargeNumbers#Estimating_e_using_SOCR_simulation