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import numpy as np

def comp_theta_mle(d):

"""

Computes the Maximum Likelihood Estimate for a given 1D training

dataset for a Rayleigh distribution.

"""

theta = len(d) / sum([x**2 for x in d])

return theta

def likelihood_ray(x, theta):

"""

Computes the class-conditional probability for an univariate

Rayleigh distribution

"""

return 2*theta*x*np.exp(-theta*(x**2))

if __name__ == "__main__":

training_data = [10, 18, 19, 22, 24, 29, 33, 40, 68]

theta = comp_theta_mle(training_data)

# Plot Probability Density Function

from matplotlib import pyplot as plt

x_range = np.arange(0, 20, 0.1)

y_range = [likelihood_ray(theta, x) for x in x_range]

plt.figure(figsize=(10,8))

plt.plot(x_range, y_range, lw=2)

plt.title('Probability density function for the Rayleigh distribution')

plt.ylabel('p(x)')

plt.xlabel('random variable x')

plt.show()