import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
from scipy.optimize import minimize

# Generate some data points that are normally distributed
np.random.seed(0) # For reproducibility
true_mu, true_sigma = 0, 0.1 # True parameters
data = np.random.normal(true_mu, true_sigma, 1000)

# Define the likelihood function
def negative_log_likelihood(params):
    # Our parameters to estimate are the mean and standard deviation
    mu, sigma = params[0], params[1]
    
    # We use the negative log likelihood because we will be minimizing the function
    # The constant term (log(2*pi)) is omitted since it doesn't affect the result
    nll = -np.sum(norm.logpdf(data, loc=mu, scale=sigma))
    return nll

# Initial guesses for mu and sigma
initial_params = [0, 1]

# Minimize the negative log likelihood
result = minimize(negative_log_likelihood, initial_params, method='nelder-mead')

# Estimated parameters
mu_est, sigma_est = result.x
print(f"Estimated mu: {mu_est}, Estimated sigma: {sigma_est}")

# Plot the histogram of the data
plt.hist(data, bins=30, density=True, alpha=0.5, label='Data histogram')

# Plot the PDF of the normal distribution with the estimated parameters
x = np.linspace(min(data), max(data), 100)
plt.plot(x, norm.pdf(x, mu_est, sigma_est), 'r-', label='Fitted normal distribution')

# Show the plot with legend
plt.legend()
plt.show()