Law Of Large Number

Law of large numbers has been defined as one among many theorems which expresses one idea. The idea expresses that with the increase in the random procedure’s numerous trials, the % difference in between the actual and expected values tend to go to "0".

In the theory of probability, LLN – Law of large numbers, has been defined as that theorem which describes the outcomes of doing same-experiment numerous times. In accordance to this law, the result’s average derived from numerous trials must be near to the value expected, & will incline to come nearer with the performance of more number of trials.

Say for e.g. a die is having six sides and has numbers 1, 2, 3, 4, 5 and 6 on each side. Now, this dice’s one roll will have same probability for producing any of the above number. Thus, dice’s one roll’s expected value is as follows:

(1+2+3+4+5+6)/6 = 3.5

As stated in the law of large numbers, when a dice is rolled numerous times, the median of the number’s values (often called as sample mean) seems to-be very near 3.5, along with an increment in accuracy when more number of dice is being rolled.

Forms of law of large numbers

There are two forms of law of large numbers – strong law and weak law. Both the versions state a common thing, i.e. with practical certainty, sample average

Converges or meets with expected value

Where X₁, X_{2  , ……} are i.i.d. (independent and identically distributed) integral-random-variable’s infinite series of number with the expected-value, i.e. E(X₁) = E(X₂) = ...= µ. By the term integral ability we mean that the variable E (|X|₁) = E (|X|₂) = ...< ∞.

Assumptions of the finite-variance, i.e. Var (X₁) =Var (X₂) = ... = σ² < ∞ aren’t necessary. Infinite or large variance makes convergence or overlapping slower, however, law of large numbers holds them anyway. Such assumptions are used often as it helps in making the evidences or proofs shorter and easier.

Differences in between weak and strong versions are related with ways of the convergence which are asserted.

These two versions are described in brief below:

• Strong Law – It is stated by weak law of large numbers that sample-averages meet in the probability in the direction of the expected-value.

  

• Weak Law – It is stated by weak law of large numbers that sample-averages meet nearly surely in the direction of the expected-value.

  

Importance of LLN

Law of large numbers has been considered important as it gives a guarantee of unchanging long-term final result for the random events. One should always remember that LLN applies only when he takes into consideration, large no. of observations.