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10
source/statistics/bayes_theorem.rst
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@ -1,7 +1,7 @@
Bayes' Theorem
---------------
This page is inpired from:
This page is inspired from:
* `MathIsFun <https://www.mathsisfun.com/data/bayes-theorem.html>`__
* `Wikipedia <https://en.wikipedia.org/wiki/Bayes%27_theorem>`__
@ -11,8 +11,8 @@ Theorem
The Bayes's theorem describes the probability of an event, based on prior knowledge of conditions
that might be related to the event.
To compute :math:`P(A|B)`, in other words, the probability of the event :math:`A` to happen, knowing that :math:`B`
already happen, the following drawing can be used:
To compute :math:`P(A|B)`, in other words, the probability of the event :math:`A` to happend, knowing that :math:`B`
already happend, the following drawing can be used:
.. image:: figures/bayes_theorem.svg
:align: center
@ -20,7 +20,7 @@ already happen, the following drawing can be used:
|
The red area represents :math:`P(B)=a3+a4`. Thus, to compute :math:`P(A|B)` we do:
The red area represents :math:`P(B)=a3+a4`. To compute :math:`P(A|B)` we do:
.. math::
P(A|B)=\frac{a3}{a3+a4}=\frac{P(A)P(B|A)}{P(A)P(B|A)+P(\neg A)P(B|\neg A)}=\frac{P(A)P(B|A)}{P(B)}
@ -68,4 +68,4 @@ Let's draw the little diagram:
&=\frac{0.01 \times 0.80}{0.01 \times 0.80 + 0.99 \times 0.10} = 0.0747
The chances that Anna really has Allergy is about 7%.
The chances that Anna really has allergy is about 7%.

7
source/statistics/bessel_correction.rst
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@ -8,7 +8,8 @@ Bessel's Correction
Bessel's correction is the use of :math:`n-1` instead of :math:`n` in the formulas for sample
variance and sample standard deviation.
In fact, using :math:`n` as a denominator leads to a biased estimator.
This variance estimator is noted :math:`s^2_n`.
The biased estimator for the sample standard deviation is noted :math:`s_n`.
The biased estimator for the sample variance is noted :math:`s^2_n`.
Lets compute the discrepency between population variance and the biased sample variance:
.. math::
@ -33,9 +34,9 @@ Lets compute the discrepency between population variance and the biased sample v
&= \frac{\sigma^2}{n}
This result shows us that the discrepency between the population and sample variance is :math:`\frac{\sigma^2}{n}`.
This result shows that the discrepency between the population and sample variance is :math:`\frac{\sigma^2}{n}`.
It is simply, the :ref:`Standard Error of the Mean <SEM>`.
From this result we can deduce how :math:`S_n^2` must be adjusted:
From this result we can deduce how :math:`s_n^2` must be adjusted to be unbiased:
.. math::
\mathbb{E} \left[ s^2_n \right] = \sigma^2 - \frac{\sigma^2}{n} = \frac{n-1}{n} \sigma^2

4
source/statistics/probability_distribution_functions.rst
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@ -6,7 +6,7 @@ Probability Distribution Functions
Probability Density Function
=============================
The Probability Density Function (PDF) is function defined for a random variable :math:`X` such that:
The Probability Density Function (PDF) is a function defined for a random variable :math:`X` such that:
.. math::
\forall (a,b) \in \mathbb{R}^2,\, P(a \le X \le b) = \int_a^b f_X(x)dx
@ -16,7 +16,7 @@ Properties:
#. :math:`\int_{-\infty}^{+\infty} f_X(x)dx=1`
#. :math:`P(X=a)=\int_{a}^{a} f_X(x)dx=0`
From property *2* it can be derived that (`source <http://yallouz.arie.free.fr/terminale_cours/probascont/prob-continue.php>`__):
From *property 2*, it can be derived that (`source <http://yallouz.arie.free.fr/terminale_cours/probascont/prob-continue.php>`__):
.. math::
P(a \le X \le b) &= P(a < X \le b) =P(a \le X < b) =P(a < X < b)