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30 lines
1.3 KiB
ReStructuredText
.. _methods_statistics:
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==========
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Statistics
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==========
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As was discussed briefly in :ref:`methods_introduction`, any given result from a
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Monte Carlo calculation, colloquially known as a "tally", represents an estimate
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of the mean of some random variable of interest. This random variable typically
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corresponds to some physical quantity like a reaction rate, a net current across
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some surface, or the neutron flux in a region. Given that all tallies are
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produced by a stochastic process, there is an associated uncertainty with each
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value reported. It is important to understand how the uncertainty is calculated
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and what it tells us about our results. To that end, we will introduce a number
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of theorems and results from statistics that should shed some light on the
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interpretation of uncertainties.
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--------------------
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Law of Large Numbers
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--------------------
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Let :math:`X_1, X_2, \dots, X_n` be an infinite sequence of independent,
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identically-distributed random variables with expected values :math:`E(X_1) =
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E(X_2) = \mu`. The sample mean :math:`\bar{X_n} = \frac{X_1 + \dots + X_n}{n}`
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converges in probability to the true mean, i.e. for all :math:`\epsilon > 0`
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.. math::
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\lim\limits_{n\rightarrow\infty} P \left ( \left | \bar{X}_n - \mu \right |
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\ge \epsilon \right ) = 0.
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