Weak law of large numbers

Weak law of large numbers

The weak law of large numbers is a result in probability theory also known as Bernoulli's theorem. Let P be a sequence of independent and identically distributed random variables, each having a mean and standard deviation.

Formula

0=limn→∞P{|X−μ|>1n} =P{limn→∞{|X−μ|>1n}} =P{X≠μ}$$\begin{gathered} 0=\underset{n\to \mathrm{\infty }}{lim}P\{|X-\mu |>\frac{1}{n}\} \\ =P\{\underset{n\to \mathrm{\infty }}{lim}\{|X-\mu |>\frac{1}{n}\}\} \\ =P\{X\ne \mu \} \end{gathered}$$

Where −

• n$n$ = Number of samples
• X$X$ = Sample value
• μ$\mu$ = Sample mean

Example

Problem Statement:

A six sided die is rolled large number of times. Figure the sample mean of their values.

Solution:

Sample Mean Calculation

Sample Mean=1+2+3+4+5+66 =216,=3.5