Statistics - Weak Law of Large Numbers

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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

$${ 0 = \lim_{n\to \infty} P \{\lvert X - \mu \rvert \gt \frac{1}{n} \} \\[7pt] \ = P \{ \lim_{n\to \infty} \{ \lvert X - \mu \rvert \gt \frac{1}{n} \} \} \\[7pt] \ = P \{ X \ne \mu \} }$$

Where −

  • ${n}$ = Number of samples

  • ${X}$ = Sample value

  • ${\mu}$ = 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 = \frac{1+2+3+4+5+6}{6} \\[7pt] \ = \frac{21}{6}, \\[7pt] \, = 3.5 }$
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