document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209758>Question 1209758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Provide a 96% confidence interval for 𝐼 = ∫₋∞^∞ e^(−x²)|sin x| dx. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850192 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850192\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Unfortunately, finding an exact analytical solution for the integral 𝐼 = ∫₋∞^∞ e^(−x²)|sin x| dx is quite challenging. Therefore, we'll need to use numerical methods and statistical techniques to estimate the confidence interval.\r<BR>\n" );
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document.write( "**1. Numerical Estimation of the Integral:**\r<BR>\n" );
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document.write( "We'll use numerical integration (e.g., Monte Carlo integration) to approximate the value of 𝐼.\r<BR>\n" );
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document.write( "Here's how we can use Monte Carlo integration:\r<BR>\n" );
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document.write( "* Generate a large number of random samples (x_i) from a suitable distribution (e.g., a normal distribution with a large standard deviation).<BR>\n" );
document.write( "* Calculate the function values e^(-x_i^2) |sin(x_i)| for each sample.<BR>\n" );
document.write( "* Estimate the integral by taking the average of these function values.<BR>\n" );
document.write( "* Repeat the process multiple times to get a sample of integral estimates.\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "import scipy.integrate as integrate\r<BR>\n" );
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document.write( "def integrand(x):<BR>\n" );
document.write( "    return np.exp(-x**2) * np.abs(np.sin(x))\r<BR>\n" );
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document.write( "# Numerical integration using scipy.integrate.quad<BR>\n" );
document.write( "result, error = integrate.quad(integrand, -np.inf, np.inf)<BR>\n" );
document.write( "print(f\"Numerical integration result: {result}\")\r<BR>\n" );
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document.write( "#Monte Carlo integration<BR>\n" );
document.write( "def monte_carlo_integral(num_samples):<BR>\n" );
document.write( "    samples = np.random.normal(0, 5, num_samples) #adjust standard deviation as needed<BR>\n" );
document.write( "    function_values = integrand(samples)<BR>\n" );
document.write( "    return np.mean(function_values)*np.sqrt(2*np.pi)*5 #adjust multiplier based on standard deviation.\r<BR>\n" );
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document.write( "num_iterations = 1000<BR>\n" );
document.write( "num_samples = 10000<BR>\n" );
document.write( "estimates = [monte_carlo_integral(num_samples) for _ in range(num_iterations)]\r<BR>\n" );
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document.write( "mean_estimate = np.mean(estimates)<BR>\n" );
document.write( "std_estimate = np.std(estimates)\r<BR>\n" );
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document.write( "print(f\"Monte Carlo mean estimate: {mean_estimate}\")<BR>\n" );
document.write( "print(f\"Monte Carlo standard deviation estimate: {std_estimate}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "**2. Statistical Analysis:**\r<BR>\n" );
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document.write( "* We now have a sample of estimates for 𝐼.<BR>\n" );
document.write( "* We can use this sample to calculate a confidence interval.<BR>\n" );
document.write( "* Since we have a reasonably large number of iterations (1000), we can use the normal distribution to approximate the confidence interval.\r<BR>\n" );
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document.write( "**3. Calculate the Confidence Interval:**\r<BR>\n" );
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document.write( "* Confidence level: 96%<BR>\n" );
document.write( "* Z-score for 96% confidence:<BR>\n" );
document.write( "    * Find the z-score corresponding to (1 + 0.96) / 2 = 0.98.<BR>\n" );
document.write( "    * Using a z-table or calculator, the z-score is approximately 2.05.<BR>\n" );
document.write( "* Margin of error (E):<BR>\n" );
document.write( "    * E = z * (std_estimate / √num_iterations)<BR>\n" );
document.write( "    * E = 2.05 * (std_estimate / √1000)<BR>\n" );
document.write( "* Confidence interval:<BR>\n" );
document.write( "    * (mean_estimate - E, mean_estimate + E)\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "z_score = 2.05<BR>\n" );
document.write( "margin_of_error = z_score * (std_estimate / np.sqrt(num_iterations))<BR>\n" );
document.write( "confidence_interval = (mean_estimate - margin_of_error, mean_estimate + margin_of_error)\r<BR>\n" );
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document.write( "print(f\"96% Confidence Interval: {confidence_interval}\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Important Notes:**\r<BR>\n" );
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document.write( "* The accuracy of the Monte Carlo integration depends on the number of samples and iterations.<BR>\n" );
document.write( "* The choice of the sampling distribution can also affect the results.<BR>\n" );
document.write( "* The standard deviation of the normal distribution used in sampling can be adjusted to increase the accuracy.<BR>\n" );
document.write( "* The results will be an approximation.\r<BR>\n" );
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document.write( "**Conceptual Outline:**\r<BR>\n" );
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document.write( "1.  **Numerical Integration:** Use a method (e.g., Monte Carlo) to get a sample of integral estimations.<BR>\n" );
document.write( "2.  **Sample Statistics:** Calculate the mean and standard deviation of those estimations.<BR>\n" );
document.write( "3.  **Z-Score:** Find the z-score for a 96% confidence level.<BR>\n" );
document.write( "4.  **Margin of Error:** Calculate the margin of error using the z-score, standard deviation, and number of iterations.<BR>\n" );
document.write( "5.  **Confidence Interval:** Construct the confidence interval using the mean estimate and margin of error.\r<BR>\n" );
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document.write( "By running the provided python code, you will get the approximated numerical value of the integral and the 96% confidence interval.<BR>\n" );
document.write( " </SPAN>\n" );
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