A uniform random variable has constant probability density across a specified interval and zero density outside it. In Monte Carlo estimation, X and Y are sampled from Unif(−R, R). section 2.2
The distribution serves two distinct computational roles:
- It supplies random points for Monte Carlo integration, where an indicator identifies samples inside a circle. section 2.2
- It specifies error and scale hyperparameters in hierarchical Bayesian regression. section 6.3
Estimating π with uniform samples
Draw X and Y independently from Unif(−R, R), test whether X² + Y² < R², average the rectangle-area contributions, and divide by R². Increasing N reduces the standard error. section 2.2
Listing 2.1 Pi estimator
import numpy as np
import matplotlib.pyplot as plt
np.random.seed(42) #1
def pi_est(radius=1, num_iter=int(1e4)): #2
X = np.random.uniform(-radius,+radius,num_iter)
Y = np.random.uniform(-radius,+radius,num_iter)
R2 = X**2 + Y**2
inside = R2 < radius**2
outside = ~inside
samples = (2*radius)*(2*radius)*inside
I_hat = np.mean(samples)
pi_hat = I_hat/radius ** 2 #3
pi_hat_se = np.std(samples)/np.sqrt(num_iter) #4
print("pi est: {} +/- {:f}".format(pi_hat, pi_hat_se))
plt.figure()
plt.scatter(X[inside],Y[inside], c='b', alpha=0.5)
plt.scatter(X[outside],Y[outside], c='r', alpha=0.5)
plt.show()
if __name__ == "__main__":
pi_est()
The implementation shows NumPy uniform sampling, circle-membership tests, empirical integration, standard-error estimation, and accepted-versus-rejected sample plots. section 2.2
