LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Monte Carlo Methods for IntegrationA Project for Math 489/889Steven R. DunbarNovember 2010Suppose we wish to numerically evaluate 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 reliably to some specified error. We will evaluate this integral with basic calculus. We will also pretend that this is an integral which is too complicated to solve analytically and evaluate it with the Monte Carlo method. Then we can compare the results to understand the error of the Monte Carlo aproach. We noticethat we could consider 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 that E[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 the mean of 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 where X is a uniformly distributed random variable on [0, \317\200/2]. Therefore J = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIyRidGOEY4LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZGLyUpYmV2ZWxsZWRHRjdGOA== E[ cos(X) ].
But we have another way to evaluate the mean and therefore the integral J, 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JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn [0, \317\200/2]. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Analytic Investigation of the Integral, using Calculus</Text-field>PkkjbXVHNiItSSRpbnRHRiQ2JCwkKiYtSSRjb3NHRiQ2I0kieEdGJCIiIkkjUGlHJSpwcm90ZWN0ZWRHISIiIiIjL0YtOyIiISwkRi8jRi5GMg==print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNj9RJyZyYXJyO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR0Y3LyUqdW5kZXJsaW5lR0Y3LyUqc3Vic2NyaXB0R0Y3LyUsc3VwZXJzY3JpcHRHRjcvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGNy8lK2V4ZWN1dGFibGVHRjcvJSlyZWFkb25seUdGNy8lKWNvbXBvc2VkR0Y3LyUqY29udmVydGVkR0Y3LyUraW1zZWxlY3RlZEdGNy8lLHBsYWNlaG9sZGVyR0Y3LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y3LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmdvLUkmbXRleHRHRiQ2NlEsYXR+NX5kaWdpdHNGJ0YvRjJGNUY4RjpGPEY+RkBGQ0ZGRkhGSkZMRk5GUEZSRlRGVkZZRmNvLSZJJmV2YWxmRyUqcHJvdGVjdGVkRzYjIiImNiMsJCokSSNQaUdGJSEiIiIiIw==Pkktc2lnbWFTcXVhcmVkRzYiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQsJComLCYtSSRjb3NHRic2I0kieEdGJCIiIkkjbXVHRiQhIiIiIiNJI1BpR0YoRjRGNS9GMTsiIiEsJEY2I0YyRjU=print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEsYXR+NX5kaWdpdHNGJ0YvRjJGQw==LSZJJmV2YWxmRyUqcHJvdGVjdGVkRzYjIiImNiMsJComLCYqJEkjUGlHRiUiIiMiIiIhIilGL0YvRi0hIiMjRi9GLg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Symbolic expression of the mean and the variance of <Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiSkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiSkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn</Equation></Text-field>The symbolic expression of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is:
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
Likewise the symbolic expression of the variance is:
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Justification of normality</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=Because LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is the sum of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= independent random variables, then by the Central Limit Theorem, we expect that for large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=KiYsJiZJIkpHNiI2I0kibkdGJiIiIiZJIkVHRiY2I0YkISIiRiktSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMtSSRWYXJHRiZGLEYtis normally distributed with mean 0 and variance 1.
Then using the results derived above, we expect that
KiYsJiZJIkpHNiI2I0kibkdGJiIiIkYlISIiRiktSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMtSSRWYXJHRiY2I0YkRio= is approximately normally distibuted with mean 0 and variance 1.
JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Finding the confidence interval values</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Therefore asking that 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 the same as requiring z > 2.57582. We can evaluate this with a little examination of the symmetric normal probability distribution function and Maple numerics:
QyQtSSV3aXRoRzYiNiNJK1N0YXRpc3RpY3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI=print(); # input placeholderLUkpUXVhbnRpbGVHNiI2JC1JJ05vcm1hbEdGJDYkIiIhIiIiJCIkJioqISIkprint(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Using the confidence interval to estimate the required number of trials</Text-field>
We desire 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 with probability greater than 0.99. This would be equivalent to finding LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= so large that
MzItSSRhYnNHJSpwcm90ZWN0ZWRHNiMqJiwmJkkiSkc2IjYjSSJuR0YsIiIiRishIiJGLy1JJXNxcnRHNiRGJkkoX3N5c2xpYkdGLDYjLUkkVmFyR0YsNiNGKkYwLCQqJC1GMjYjLCQqKEkjUGlHRiYiIiNJJnNpZ21hR0YsRkBGLkYwI0YvIiIlRjAkIiImISIjL0Y5LCQqJC1GMjYjLUkiKkdGJjYkLCQqJEY/RkBGQiwkKigsJkZQRi8hIilGL0YvRj9GRkYuRjAjRi9GQEYwRkQ=LUkiPkclKnByb3RlY3RlZEc2JCwkKiQtSSVzcXJ0RzYkRiRJKF9zeXNsaWJHNiI2Iy1JIipHRiQ2JCwkKiRJI1BpR0YkIiIjIyIiIiIiJSwkKigsJkYyRjYhIilGNkY2RjMhIiNJIm5HRiwhIiIjRjZGNEY+JCIiJkY8JCInI2VkIyEiJg==LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEsc29sdmV+Zm9yfm5GJ0YvRjJGQw==LUkmc29sdmVHNiI2JDwjMiQiJyNlZCMhIiYsJComIiIjIyIiIkYtKiYsJiokSSNQaUclKnByb3RlY3RlZEdGLUYvISIpRi9GL0kibkdGJCEiIiNGN0YtJCIjNSEiIzcjRjY=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=So if we choose more than 621 samples, we should obtain an approximation that has an error of less than 0.05 with probability greatly than 0.99.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Experiment to determine J</Text-field>PkkiWEc2Ii1JKFVuaWZvcm1HRiQ2JCIiISwkSSNQaUclKnByb3RlY3RlZEcjIiIiIiIjprint(); # input placeholderQyQ+SSVzYW1wRzYiLUknU2FtcGxlR0YlNiRJIlhHRiUiJEAnIiIiprint(); # input placeholderQyQ+SSpjb3NTYW1wbGVHNiItSSRtYXBHJSpwcm90ZWN0ZWRHNiRJJGNvc0c2JEYoSShfc3lzbGliR0YlSSVzYW1wR0YlISIi%;Pkk4bW9udGVjYXJsb0FwcHJveGltYXRpb25HNiItSSVNZWFuR0YkNiNJKmNvc1NhbXBsZUdGJA==print(); # input placeholderPkkuYWJzb2x1dGVFcnJvckc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUkkYWJzR0YnNiMsJiomSSNQaUdGJyIiIkkjbXVHRiRGLyNGLyIiIyomRi5GL0k4bW9udGVjYXJsb0FwcHJveGltYXRpb25HRiRGLyMhIiJGMg==print(); # input placeholderOur deliberations above tell us we could repeat this experiment 100 times and expect to get a similar result at least 99 times.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Experiment to determine the integral of sine</Text-field>We believe that 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 should be very similar to 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 and the distribution of 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 should be similar to 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= uniformly distributed on 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.That means our inution tells us that we should use a sample size of 621 or greater.
QyQ+SSVzYW1wRzYiLUknU2FtcGxlR0YlNiRJIlhHRiUiJEAnIiIiprint(); # input placeholderQyU+SSpzaW5TYW1wbGVHNiItSSRtYXBHJSpwcm90ZWN0ZWRHNiRJJHNpbkc2JEYoSShfc3lzbGliR0YlSSVzYW1wR0YlISIiPkk8bW9udGVjYXJsb0FwcHJveGltYXRpb25TaW5lR0YlLCQqJkkjUGlHRigiIiItSSVNZWFuR0YlNiNGJEY0I0Y0IiIjprint(); # input 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LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSRzaW5HRiQ2I0kieEdGJy9GLDsiIiEsJEkjUGlHRiUjIiIiIiIjprint(); # input placeholderJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRitGLw==