File:Law of large numbers.gif

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Summary

DescriptionLaw of large numbers.gif	English: A plot of the average of n simulated independent Bernoulli trials (each taking a value of +/- 1) to show the law of large numbers and the law of the iterated logorithm. The graphical difficulty here is that we want to show details for very large values of n. If the y-axis were left untransformed, all we could see is that it converges to zero. If we used a log transformation it would be upset with the negative values. So the y is transformed by something that is very close to a log transformation, namely, y^.12. (The .12 looks nice. Other small powers look about the same.) Humans are very good at seeing "straight lines". So if we transform the x-axis to be almost log also we can make the upper and lower envelopes look almost straight. The correct power is -1/2 of the power used for the y axis. So the plot is x^-.06 vs y^.12. These are basically "log(x)" vs "log(y)". But by using these values, the natural square root behaviour now is a straight line. If you have suggestions for improving the image, let me know and I can generate a new one. A detailed view of the right hand most "pixel" is:
Date	9 May 2009
Source	Own work
Author	Dean P Foster

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File history

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	Date/Time	Dimensions	User	Comment
current	11:53, 15 May 2009	1,000 × 600 (21 KB)	Dean P Foster	Why won't it accept this file???
	11:51, 15 May 2009	1,000 × 600 (21 KB)	Dean P Foster	Reverted to version as of 11:44, 15 May 2009. THis is the correct file
	11:49, 15 May 2009	1,000 × 600 (21 KB)	Dean P Foster	one more time! added sqrt(n)
	11:46, 15 May 2009	1,000 × 600 (21 KB)	Dean P Foster	added sqrt(n) line
	11:45, 15 May 2009	1,000 × 600 (15 KB)	Dean P Foster	oops wrong file
	11:44, 15 May 2009	1,000 × 600 (21 KB)	Dean P Foster	added sqrt(n) line
	19:55, 13 May 2009	1,000 × 600 (15 KB)	Dean P Foster	{{Information \|Description={{en\|1=A plot of the average of +/-1 bernulli trials to show the law of large numbers and the law of the iterated logorithm. }} \|Source=Own work by uploader \|Author=Dean P Foster \|Date= 2009-05-09 \|Permis