Computing a Confidence Interval for a Gaussian-Weighted Integral

Question

Requirement: Provide a 96% confidence interval for .

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I used numerical integration (e.g., Monte Carlo integration) to approximate the value of I.

Here is how I could use Monte Carlo integration:

• Generate a large number of random samples (x_i) from a suitable distribution (e.g., a normal distribution with a large standard deviation).
• Calculate the function values for each sample.
• Estimate the integral by taking the average of these function values.
• Repeat the process multiple times to get a sample of integral estimates.

import numpy as np
import scipy.integrate as integrate

def integrand(x):
    return np.exp(-x**2) * np.abs(np.sin(x))

# Numerical integration using scipy.integrate.quad
result, error = integrate.quad(integrand, -np.inf, np.inf)
print(f"Numerical integration result: {result}")

# Monte Carlo integration
def monte_carlo_integral(num_samples):
    samples = np.random.normal(0, 5, num_samples) # adjust standard deviation as needed
    function_values = integrand(samples)
    return np.mean(function_values)*np.sqrt(2*np.pi)*5 # adjust multiplier based on standard deviation.

num_iterations = 1000
num_samples = 10000
estimates = [monte_carlo_integral(num_samples) for i in range(num_iterations)]

mean_estimate = np.mean(estimates)
std_estimate = np.std(estimates)

print(f"Monte Carlo mean estimate: {mean_estimate}")
print(f"Monte Carlo standard deviation estimate: {std_estimate}")

• Currently, I had a sample of estimates for I.
• I could use this sample to calculate a confidence interval.
• Since I had a reasonably large number of iterations (1000), I could use the normal distribution to approximate the confidence interval.
• Confidence level: 96%.
• z-score for 96% confidence:
  • Find the z-score corresponding to (1 + 0.96) / 2 = 0.98.
  • Using a z-table or calculator, the z-score is approximately 2.05.
• Margin of error (E):
• Confidence interval: (mean_estimate - E, mean_estimate + E)

z_score = 2.05
margin_of_error = z_score * (std_estimate / np.sqrt(num_iterations))
confidence_interval = (mean_estimate - margin_of_error, mean_estimate + margin_of_error)

print(f"96% Confidence Interval: {confidence_interval}")

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At the moment, I don't know how to solve this problem mathematically, so I hope anyone can support me to analyze it.