The weak law of large numbers (cf. the strong law of large numbers ) is a result in probability theory also known as Bernoulli's
theorem. Let ,
...,
be a sequence of independent and identically distributed random variables, each having
a mean and standard
deviation .
Define a new variable
(1)
Then, as ,
the sample mean
equals the population mean of each variable.
In addition,
Therefore, by the Chebyshev inequality , for all ,
(10)
As ,
it then follows that
(11)
(Khinchin 1929). Stated another way, the probability that the average for an arbitrary positive quantity
approaches 1 as
(Feller 1968, pp. 228-229).
See also Asymptotic Equipartition Property ,
Central Limit Theorem ,
Chebyshev
Inequality ,
Frivolous Theorem of
Arithmetic ,
Law of Truly Large Numbers ,
Strong Law of Large Numbers
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References Feller, W. "Laws of Large Numbers." Ch. 10 in An
Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed.
New York: Wiley, pp. 228-247, 1968. Feller, W. "Law of Large
Numbers for Identically Distributed Variables." §7.7 in An
Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed.
New York: Wiley, pp. 231-234, 1971. Khinchin, A. "Sur la loi
des grands nombres." Comptes rendus de l'Académie des Sciences 189 ,
477-479, 1929. Papoulis, A. Probability,
Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill,
pp. 69-71, 1984. Referenced on Wolfram|Alpha Weak Law of Large Numbers
Cite this as:
Weisstein, Eric W. "Weak Law of Large Numbers."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/WeakLawofLargeNumbers.html
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