Wednesday, December 4, 2013

Law of large numbers


  • Definition
    • a theorem that describes
      the average of the results obtained from a large number of trials
      should be close to the expected value      ,
      and
      will tend to become closer as more trials are performed.
    • the weak law
      \lim_{n\to\infty}\Pr\!\left(\,|\overline{X}_n-\mu| > \varepsilon\,\right) = 0.
    • the strong law
      \Pr\!\left( \lim_{n\to\infty}\overline{X}_n = \mu \right) = 1.
  • Example
    • An illustration of the law of large numbers using a particular run of rolls of a single die.
  • Comments
    • the above picture describes intuitively 'law of large numbers'
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