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Published in:
2014 | OriginalPaper | Chapter
5. Moments and Laws of Large Numbers
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Abstract
The most important characteristic quantities of random variables are the median, expectation and variance. For large n, the expectation describes the typical approximate value of the arithmetic mean (X
1+…+X
n
)/n of independent and identically distributed random variables (law of large numbers).
In this chapter, we first show the weak law of large numbers and then present Etemadi’s strong law of large numbers. As an example we study the average length of a random message (source coding theorem). Under additional moment conditions, we investigate the speed of convergence in the law of large numbers. Finally, as an important example, we consider the Poisson process.