Errors in Sampling

We reuse the setup from the computation of $\pi$, but run this many times for a number of sample counts.

import numpy as np
import matplotlib.pyplot as plt

sample_counts = np.arange(10, 2000, 20)

nexperiments = 1000

avg_errors = []

for nsamples in sample_counts:
   
    errors = []
    
    for experiment in range(nexperiments):
        # draw samples
        samples = np.random.rand(2, nsamples)
        
        # compute p
        p = np.zeros(nsamples)
        r = samples[0]**2 + samples[1]**2
        p[r<=1] = 1
        
        approx_pi = 4*np.sum(p) / nsamples
        errors.append(np.abs(approx_pi - np.pi))
        
    avg_errors.append(np.sum(errors)/nexperiments)
    
avg_errors = np.array(avg_errors)

plt.loglog(sample_counts, avg_errors)
plt.xlabel("# Samples")
plt.ylabel("Error")
plt.grid()

Empirically: Average error

$$E=O(\frac1{\sqrt n})$$

Precise statement: Central Limit Theorem