computing integrals draw ** random points in the rectangle np random uniform( ny np random uniform( ncompute sum of values inside the integration domain f_mean num_inside number of , points inside domain ( >= for in range(len( ))for in range(len( ))if ( [ ] [ ]> num_inside + f_mean + ( [ ] [ ]f_mean f_mean/float(num_insidearea num_inside/float( ** )*( )*( return area*f_mean (see the file mc_double py verification simple test case is to check the area of rectangle oe oe : embedded in rectangle oe oe the right answer is but monte carlo integration isunfortunatelynever exact so it is impossible to predict the output of the algorithm all we know is that the estimated integral should approach as the number of random points goes to infinity alsofor fixed number of pointswe can run the algorithm several times and get different numbers that fluctuate around the exact valuesince different sample points are used in different calls to the monte carlo integration algorithm : the area of the rectangle can be computed by the integral dydxso in this case we identify xyd and the function can be specified as ( if xyis inside oe oe : and otherwise here is an example on how we can utilize the montecarlo_double function to compute the area for different number of samplesfrom mc_double import montecarlo_double def (xy)return ( if ( < < and < < else - montecarlo_double(lambda xy montecarlo_double(lambda xy montecarlo_double(lambda xy montecarlo_double(lambda xy montecarlo_double(lambda xy montecarlo_double(lambda xy we see that the values fluctuate around fact that supports correct implementationbut in principlebugs could be hidden behind the inaccurate answers it is mathematically known that the standard deviation of the monte carlo estimate of an integral converges as = where is the number of samples this