72 lines
2 KiB
ReStructuredText
72 lines
2 KiB
ReStructuredText
Notations
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---------------
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This page was inspired from the following `Wikipedia page <https://en.wikipedia.org/wiki/Notation_in_probability_and_statistics>`_.
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Sample and Population
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=====================
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A *population* is the entire population that you want to analyze. It is a exhaustive set. A *sample* is a subset of the population. It is a non-exhaustive set.
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.. figure:: figures/population_sample.svg
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:align: center
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:width: 300px
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*Inferential statistics* allows you draw conclusions (with a certain degree of confidence) on a population using samples.
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A Tale of Schizophrenia
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================================
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Two different notation conventions are used. The one to use depends if you are working on a *population* or a *sample*:
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.. list-table:: Notation
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:align: center
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:header-rows: 1
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* - Metric
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- Population
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- Sample
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- Notes
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* - Sample mean
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- :math:`\mu`
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- :math:`\overline{x}`
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-
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* - Variance
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- :math:`\sigma^2`
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- :math:`s^2`
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- :math:`s^2_n` without `Bessel's Correction <bessel_correction>`__
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* - Standard deviation
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- :math:`\sigma`
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- :math:`s`
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- :math:`s_n` without `Bessel's Correction <bessel_correction>`__
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To determine the metric of a population (say :math:`\mu`) using a sample, we use an estimator.
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In the case of :math:`\mu`, we use :math:`\overline{x}` as an estimator.
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.. note::
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Estimators can also be denoted with the hat symbol.
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For example :math:`\hat{\mu}\equiv\overline{x}`.
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Operators
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==========
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Expected value
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^^^^^^^^^^^^^^^
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The expected value (*espérance*) of a random variable :math:`X` is noted :math:`\mathbb{E}[X]`.
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It as the following linearity properties:
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.. math::
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\mathbb{E}[X+Y]=\mathbb{E}[X] + \mathbb{E}[Y]
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.. math::
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\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]
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Variance
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^^^^^^^^
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The variance operator of a random variable :math:`X` is noted :math:`\mathbb{V}[X]` or :math:`\mathrm{Var}[X]`.
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Coraviance
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^^^^^^^^^^
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The covariance operator of a random variable :math:`X` and :math:`Y`
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is noted :math:`\mathrm{Cov}[X,Y]`.
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