import random, math, pylab

def direct_pi(N): n_hits = 0 for i in range(N): x, y = random.uniform(-1.0, 1.0), random.uniform(-1.0, 1.0) if x ** 2 + y ** 2 < 1.0: n_hits += 1 return n_hits

n_runs = 500 n_trials_list = [] sigmasqs = [] for poweroftwo in range(4, 13): n_trials = 2 ** poweroftwo sigmasq = 0.0 for run in range(n_runs): pi_est = 4.0 * direct_pi(n_trials) / float(n_trials) sigmasq += (pi_est - math.pi) ** 2 sigmasqs.append(math.sqrt(sigmasq/(n_runs))) n_trials_list.append(n_trials)

pylab.plot(n_trials_list, sigmasqs, 'o') pylab.xscale('log') pylab.yscale('log') pylab.xlabel('number of trials') pylab.ylabel('root mean square deviation') pylab.title('Direct sampling of pi: root mean square deviation vs. n_trials') pylab.savefig('direct_sampling_rms_deviation.png') pylab.show()