Weak Law of Large Numbers


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_{nto infty} P {lvert X – mu rvert gt frac{1}{n} } \[7pt]
= P { lim_{nto 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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