From bf2abddbb9e92bbb9bb58f457ebd735c8e21299a Mon Sep 17 00:00:00 2001 From: Loic Guegan Date: Wed, 18 Oct 2023 22:09:21 +0200 Subject: [PATCH] Minor changes --- source/statistics/bayes_theorem.rst | 10 +++++----- source/statistics/bessel_correction.rst | 7 ++++--- .../statistics/probability_distribution_functions.rst | 4 ++-- 3 files changed, 11 insertions(+), 10 deletions(-) diff --git a/source/statistics/bayes_theorem.rst b/source/statistics/bayes_theorem.rst index 4e19a3b..8d310f4 100644 --- a/source/statistics/bayes_theorem.rst +++ b/source/statistics/bayes_theorem.rst @@ -1,7 +1,7 @@ Bayes' Theorem --------------- -This page is inpired from: +This page is inspired from: * `MathIsFun `__ * `Wikipedia `__ @@ -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%. diff --git a/source/statistics/bessel_correction.rst b/source/statistics/bessel_correction.rst index 4719331..be1bbb4 100644 --- a/source/statistics/bessel_correction.rst +++ b/source/statistics/bessel_correction.rst @@ -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 `. -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 diff --git a/source/statistics/probability_distribution_functions.rst b/source/statistics/probability_distribution_functions.rst index c2425ed..370152e 100644 --- a/source/statistics/probability_distribution_functions.rst +++ b/source/statistics/probability_distribution_functions.rst @@ -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 `__): +From *property 2*, it can be derived that (`source `__): .. math:: P(a \le X \le b) &= P(a < X \le b) =P(a \le X < b) =P(a < X < b)