## Weak Law of Large Numbers

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.
 (2)

 (3)

Therefore, by the Chebyshev Inequality, for all ,
 (4)

As , it then follows that
 (5)

for arbitrarily small; i.e., as , the sample Mean is the same as the population Mean.

Stated another way, if an event occurs times in Trials and if is the probability of success in a single Trial, then the probability that for an arbitrary Positive quantity approaches 1 as .

See also Law of Truly Large Numbers, Strong Law of Large Numbers