Weak Law of Large Numbers (WLLN):

As a direct application of Chebyshav's lemma we get the weak law of Large Numbers (wlln).

Theorem 1:

Let x¯ be the sample mean of a random sample X of size n from a population with mean μ and finite variance σ², then

lim P {¦ x¯ - μ¦> ε } =0, for every ε >0

n→∞

Proof

Let X = (X₁, X₂, . X_n, ) Then Var (X_i ) = σ² for i = 1,2, ..., n and so Var (x¯) = σ²/n. Using Chcbyshev's lemma directly to the random variable x¯  we get for every ε > 0

P{¦ x¯ -μ ¦ >ε } ≤ σ²/n² ε

Hence

lim P {¦ x¯ - μ¦> ε } =0,

n→∞

The WLLN is applicable even for variables X₁,X₂,.. ., X_n, which may not be identically and independently distributed (i.e. may not be the elements of a random sample) and may not have finite variances. However, we shall not prove the general cases of WLLN here.

In particular case when each X1, X assumes either for 0 according to the success of the failure in a Bernoulli's trail along with probability of success p we can rewrite the WLLN in the following form

lim P {¦S/n-P¦>ε) = 0 for any t>0

n→∞

where S= X₁ +X₂ + ... + X_n is the total number of successes in n independent Bernoulli's trial. This result known as Bernoulli's WLLN.

Previous assertation gives a theoretical justification for the intuitive frequency definition of probability. In indicates that if in n identical trails an events A occurs S turns then as n→ x, f_A = S/n the relative frequency of event A converges in probability to p. The probability of the occurrence of A as n→ x.

Remarks:

When the law of law number is stated more precisely, it is called the strong law of large number. The basic difference arises in mode of convergence we shall not consider the strong law of large numbers in this course.