law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. www.unitedworld.edu.in

1. LAW OF LARGE NUMBERS

2. LAW OF LARGE NUMBERS
A "law of large numbers" is one of several theorems expressing the idea that as the
number of trials of a random process increases, the percentage difference between the
expected and actual values goes to zero.
STRONG LAW OF LARGE NUMBERS
The sequence of variates with corresponding means obeys the strong law of large numbers if, to every
pair , there corresponds an such that there is probability or better that for every , all
inequalities
(1)
for , , ..., will be satisfied, where
(2)
(3)
(Feller 1968). Kolmogorov established that the convergence of the sequence
(4)
Sometimes called the Kolmogorov criterion, is a sufficient condition for the strong law of large numbers to apply to the
sequence of mutually independent random variables with variances (Feller 1968).
WEAK LAW OF LARGE NUMBERS
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

3. (1)
Then, as , the sample mean equals the population mean of each variable.
(2)
(3)
(4)
(5)
In addition,
(6)
(7)
(8)
(9)
Therefore, by the Chebyshev inequality, for all ,
(10)
As , it then follows that
(11)
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