.. _sphx_glr_api_gallery_probability_limit_theorems:

Limit theorems and inequalities
---------------------------------

Bernoulli's law of large numbers, the De Moivre-Laplace central limit
theorem, and the Bienaymé-Chebyshev inequality, simulated and verified.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Bernoulli&#x27;s &quot;golden theorem&quot; (1713): in n independent trials of an event with probability p, the relative frequency of the event converges to p as n grows. Here a biased coin with p = 0.3 is tossed a million times, and the running fraction of heads homes in on 0.3 while its typical error shrinks like \sqrt{p(1-p)/n}.">

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  .. image:: /api/gallery/probability/limit_theorems/images/thumb/sphx_glr_plot_01_law_of_large_numbers_thumb.png
    :alt:

  :doc:`/api/gallery/probability/limit_theorems/plot_01_law_of_large_numbers`

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      <div class="sphx-glr-thumbnail-title">Jacob Bernoulli's law of large numbers: frequencies settle down</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="De Moivre (1733) showed that the binomial distribution, suitably standardized, approaches the normal curve for large n. Laplace (1810) extended this to sums of many independent variables of almost any distribution: the standardized mean Z = (\bar X_n - \mu)/(\sigma/\sqrt n) tends to \mathcal N(0, 1), whatever the shape of the summands.">

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  .. image:: /api/gallery/probability/limit_theorems/images/thumb/sphx_glr_plot_02_central_limit_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/probability/limit_theorems/plot_02_central_limit_theorem`

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      <div class="sphx-glr-thumbnail-title">De Moivre-Laplace and the central limit theorem: the bell curve emerges</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="P(|X-\mu| \geq k\sigma) \leq 1/k^2 holds for any distribution with finite variance. Comparing it with exact tails shows how conservative a universal bound has to be.">

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  .. image:: /api/gallery/probability/limit_theorems/images/thumb/sphx_glr_plot_03_chebyshev_inequality_thumb.png
    :alt:

  :doc:`/api/gallery/probability/limit_theorems/plot_03_chebyshev_inequality`

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      <div class="sphx-glr-thumbnail-title">Chebyshev's inequality: one bound for every distribution</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/probability/limit_theorems/plot_01_law_of_large_numbers
   /api/gallery/probability/limit_theorems/plot_02_central_limit_theorem
   /api/gallery/probability/limit_theorems/plot_03_chebyshev_inequality