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zv@epVl2LlWJ1pY%2r(1r4%4N2fvhgebVe_+FJGcd+iv`(oJuKJ02bwW)FL4W<7@;1 zpsZ<U{Jxs2*>&C+8My*1dLpHk=+J(hu&u^FLPpghqC%8Tb9~Dpr^)~1)9(A1E7J~a w21$uOjx%&g%Dke_@n@6h-hX((i`vd-Gv<vhZakACnGGMg{nhn|%Y+sG0@~38zyJUM literal 0 HcmV?d00001 diff --git a/book/sandbox/continuous/other_distr.md b/book/sandbox/continuous/other_distr.md index 0a41bc2d..7f86288c 100644 --- a/book/sandbox/continuous/other_distr.md +++ b/book/sandbox/continuous/other_distr.md @@ -1,12 +1,34 @@ # Non-Gaussian distributions -In the previous section, Gaussian. +So far, you have seen that we can use a parametric distribution to model the uncertainty in our observations. Also, you have revisited one of those parametric distributions, the Gaussian distribution, which is a widely used model since it has lots of applications such as modelling natural processes or measurement errors. Here, we will discuss an interesting property of this distribution, the lack of **asymmetry**, and introduce the concept of the **tail of the distribution**. -## Concept of tail, asymmetry +## Asymmetry -Tail +Asymmetry is a relevant property of a distribution which analyzes its shape. In the Figure below, you have the PDF of a Gaussian distribution. +```{figure} /sandbox/continuous/figures/one_gaussian.png + +--- + +--- +Gaussian PDF. +``` + +As you can see, the shape of the PDF is perfectly symmetric with respect to the mode[^mode] of the distribution. This leads to an interesting property: for perfectly symmetric distributions the mode, mean[^mean] and median[^median] have the same value. + +As you can imagine, it is also possible to analyze the asymmetry of an empirical distribution. To this end, we can visually inspect the empirical PDF and compute the *sample coefficient of skewness* from a set of observations as + +$ +g_1 = \frac{\sum_{i=1}^n{(x_i-\overline{x})^3}}{ns^3} +$ + +where $x_i$ are the observations, $\overline{x}$ is the mean of the observations, $n$ is the number of observations, and $s$ is the standard deviation. + +**And what happens when my observations are not symmetric?** + +However, it is not always the case. It is very common to find asymmetric distributions +## Tail of the distribution ## Lognormal distribution @@ -19,4 +41,8 @@ for instance ## Gumbel distribution -List of distr is tentative \ No newline at end of file +List of distr is tentative + +[^mode]: The mode is the most frequently observed value and, thus, the value of the distribution with the highest value of the density in its PDF. +[^mean]: Visually from the PDF, we can define the mean as the value of the random variable which is the balance point of the distribution. +[^median]: Visually from the PDF, we can define the median as the value of the random variable which divides the PDF in two areas of the same value. \ No newline at end of file -- GitLab From 1a0bd5e291b687ed7ae816feaf5582d28eb0c61e Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Fri, 21 Jul 2023 12:58:48 +0200 Subject: [PATCH 02/21] finish symmetry and tail --- book/sandbox/continuous/other_distr.md | 25 ++++++++++++++++--------- 1 file changed, 16 insertions(+), 9 deletions(-) diff --git a/book/sandbox/continuous/other_distr.md b/book/sandbox/continuous/other_distr.md index 7f86288c..9fba3177 100644 --- a/book/sandbox/continuous/other_distr.md +++ b/book/sandbox/continuous/other_distr.md @@ -3,7 +3,7 @@ So far, you have seen that we can use a parametric distribution to model the uncertainty in our observations. Also, you have revisited one of those parametric distributions, the Gaussian distribution, which is a widely used model since it has lots of applications such as modelling natural processes or measurement errors. Here, we will discuss an interesting property of this distribution, the lack of **asymmetry**, and introduce the concept of the **tail of the distribution**. -## Asymmetry +## Concepts of asymmetry and tail of the distribution Asymmetry is a relevant property of a distribution which analyzes its shape. In the Figure below, you have the PDF of a Gaussian distribution. @@ -27,21 +27,28 @@ where $x_i$ are the observations, $\overline{x}$ is the mean of the observations **And what happens when my observations are not symmetric?** -However, it is not always the case. It is very common to find asymmetric distributions -## Tail of the distribution +In that case, Gaussian distribution may not be the most appropriate model and we need to look for a different model. Actually, it is very common to find asymmetric distributions when modelling natural processes! -## Lognormal distribution +If we go back to the wind data that we previously introduced and have a look at the PDF, we can easily notice that it is not symmetric. The mass in the PDF is not distributed equally at both sides of the mode. Actually, the shape is similar to a bell with the center around $W_s = 5m/s$, but we can see that the distribution expands towards the positive values creating what we call the **tail of the distribution**. Since the tail expands towards positive values, we say that the PDF has *positive skewness*. -for instance +```{figure} /sandbox/continuous/figures/epdf_wind.png -## Exponential distribution +--- + +--- +Empirical probability density function of the wind speed data. +``` + +Many natural phenomena present positive skewness. This means that the mode, the mean and the median do not have the same value anymore. For a positively skewed PDF: mode < median < mean. This inequality is reversed if the PDF is negatively skewed. + +**And why are asymmetry and tail that important?** -for instance +As you can imagine, asymmetry is an important criteria when selecting the parametric distribution to model the uncertainty in our observations. The shape of the parametric distribution needs to fit the shape of the empirical distribution function. +Moreover, the tail of the distribution is modelling the highest and lowest values of our variable and, typically, they are the ones we use for design. For instance, if we are designing a building, we would be interested in the high wind speeds that it has to withstand (loading conditions). Also, if we are considering the nutrients required to cultivate mussels, we would analyze the lowest values to ensure that the mussels have enough food to survive. Thus, it is important to choose a parametric distribution which is able to model the behavior in the tail when the observations are not symmetric. -## Gumbel distribution +In the following sections, we will introduce some common parametric distributions, as well as their characteristics, to give an overview of some possible models you may apply to model your data. But keep in mind that the list is not exhaustive, there are a lot of distributions! -List of distr is tentative [^mode]: The mode is the most frequently observed value and, thus, the value of the distribution with the highest value of the density in its PDF. [^mean]: Visually from the PDF, we can define the mean as the value of the random variable which is the balance point of the distribution. -- GitLab From 44d5bc11f14058a2a939588f50b174f84179ab30 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Fri, 21 Jul 2023 13:30:19 +0200 Subject: [PATCH 03/21] uniform distribution --- book/_toc.yml | 1 + book/sandbox/continuous/Uniform.md | 37 ++++++++++++++++++++ book/sandbox/continuous/figures/uniform.png | Bin 0 -> 27527 bytes 3 files changed, 38 insertions(+) create mode 100644 book/sandbox/continuous/Uniform.md create mode 100644 book/sandbox/continuous/figures/uniform.png diff --git a/book/_toc.yml b/book/_toc.yml index 66ab5224..6735b02d 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -23,6 +23,7 @@ parts: - file: sandbox/continuous/empirical - file: sandbox/continuous/Param_distr.md - file: sandbox/continuous/other_distr.md + - file: sandbox/continuous/Uniform.md - file: sandbox/continuous/Parameterization.md - file: sandbox/continuous/fitting.md - file: sandbox/continuous/GOF diff --git a/book/sandbox/continuous/Uniform.md b/book/sandbox/continuous/Uniform.md new file mode 100644 index 00000000..6300b6bc --- /dev/null +++ b/book/sandbox/continuous/Uniform.md @@ -0,0 +1,37 @@ + +# Uniform distribution + +The uniform distribution is the simplest type of continuous parametric distribution and, as it is implied in its name, the PDF has a constant value along a given interval $[a,b]$, where $a < b$. The PDF of the uniform is given by + +$ +f(x) = \frac{1}{b-a} \hspace{1cm} for \ a\leq x \leq b +$ + +$ +f(x) = 0 \hspace{1cm} otherwise +$ + +Note that all values in the distribution are between the lower limit $a$ and the higher limit $b$ and are equally likely to occur. The left pannel of the figure below, shows an example of a uniform PDF with $a=-1$ and $b=1$. + +```{figure} /sandbox/continuous/figures/uniform.png + +--- + +--- +Uniform distribution function: (left) PDF, and (right) CDF. +``` +As seen in previous chapters, the CDF is the integral of the PDF. Thus, as we integrate over a constant, the CDF presents a linear shape, as shown in the right pannel of the figure above. 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b/book/_toc.yml @@ -24,6 +24,7 @@ parts: - file: sandbox/continuous/Param_distr.md - file: sandbox/continuous/other_distr.md - file: sandbox/continuous/Uniform.md + - file: sandbox/continuous/Exponential.md - file: sandbox/continuous/Parameterization.md - file: sandbox/continuous/fitting.md - file: sandbox/continuous/GOF diff --git a/book/sandbox/continuous/Exponential.md b/book/sandbox/continuous/Exponential.md new file mode 100644 index 00000000..c5b02aef --- /dev/null +++ b/book/sandbox/continuous/Exponential.md @@ -0,0 +1,44 @@ + +# Exponential distribution + +Another widely used distribution function is the Exponential distribution. For instance, it is applied to model the waiting time between succesive events of a Poisson process. The PDF of the Exponential distribution is given by + +$ +f(x) = \lambda e^{-\lambda x} \hspace{1cm} for \ x \geq 0, \lambda>0 +$ + +$ +f(x) = 0 \hspace{1cm} otherwise +$ + +where $\lambda$ is the parameter of the distribution, which is often called *rate*. In the RIGHT pannel of the figure below, an example of two Exponential distributions with $\lambda =1$ and $\lambda = 2$ is shown. As you can see, the maximum density in the PDF of en Exponential distribution is located at zero and it is followed by an Exponential decay. The higher the parameter $\lambda$, the higher the value of the density in $x=0$ and the faster the decay. In other words, the higher the parameter $\lambda$, the more concentrated the values of the random variable which are likely to occur and, thus, the lower the standard deviation. This can be seen on the left pannel of the figure, where random samples of the distribution are plotted. There you can see how higher values of the random variable $x$ appear when $\lambda = 1$, presenting then a higher dispersion. + +```{figure} /sandbox/continuous/figures/exponential.png + +--- + +--- +Exponential distribution function: (left) random samples, and (right) PDF. +``` + + + +Integrating the PDF, the following expression of the CDF is obtained + +$ +F(x) = 1 - e^{-\lambda x} +$ + +which is visually represented in the figure below. + +```{figure} /sandbox/continuous/figures/exponential_cdf.png + +--- + +--- +Exponential distribution function: CDF. +``` + +The influence of the parameter $\lambda$ can also be observed in the CDF. The higher the parameter $\lambda$, the faster the CDF reaches values of the non-exceedance probability close to 1. Thus, lower variability of the random variable is obtained. + +**Memoryless property of the exponential distribution? Exercise with that? 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zyb<C`dVLG3FlzQkQE<NTKX0>3;-y3hC^gZM|9}22`64W-P%!_Ylzj=_FUYJj-C&Y$ H<Qo3p(9IKh literal 0 HcmV?d00001 -- GitLab From cf6eeffa245b74c3694be09cb694875cfe92ea4e Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 11:26:02 +0200 Subject: [PATCH 05/21] more exponential --- book/sandbox/continuous/Exponential.md | 31 ++++++++++++++++++++++++-- 1 file changed, 29 insertions(+), 2 deletions(-) diff --git a/book/sandbox/continuous/Exponential.md b/book/sandbox/continuous/Exponential.md index c5b02aef..389aed04 100644 --- a/book/sandbox/continuous/Exponential.md +++ b/book/sandbox/continuous/Exponential.md @@ -11,7 +11,7 @@ $ f(x) = 0 \hspace{1cm} otherwise $ -where $\lambda$ is the parameter of the distribution, which is often called *rate*. In the RIGHT pannel of the figure below, an example of two Exponential distributions with $\lambda =1$ and $\lambda = 2$ is shown. As you can see, the maximum density in the PDF of en Exponential distribution is located at zero and it is followed by an Exponential decay. The higher the parameter $\lambda$, the higher the value of the density in $x=0$ and the faster the decay. In other words, the higher the parameter $\lambda$, the more concentrated the values of the random variable which are likely to occur and, thus, the lower the standard deviation. This can be seen on the left pannel of the figure, where random samples of the distribution are plotted. There you can see how higher values of the random variable $x$ appear when $\lambda = 1$, presenting then a higher dispersion. +where $\lambda$ is the parameter of the distribution, which is often called *rate*. In the right pannel of the figure below, an example of two Exponential distributions with $\lambda =1$ and $\lambda = 2$ is shown. As you can see, the maximum density in the PDF of en Exponential distribution is located at zero and it is followed by an Exponential decay. The higher the parameter $\lambda$, the higher the value of the density in $x=0$ and the faster the decay. In other words, the higher the parameter $\lambda$, the more concentrated the values of the random variable which are likely to occur and, thus, the lower the standard deviation. This can be seen on the left pannel of the figure, where random samples of the distribution are plotted. There you can see how higher values of the random variable $x$ appear when $\lambda = 1$, presenting then a higher dispersion. ```{figure} /sandbox/continuous/figures/exponential.png @@ -41,4 +41,31 @@ Exponential distribution function: CDF. The influence of the parameter $\lambda$ can also be observed in the CDF. The higher the parameter $\lambda$, the faster the CDF reaches values of the non-exceedance probability close to 1. Thus, lower variability of the random variable is obtained. -**Memoryless property of the exponential distribution? Exercise with that? Time between earthquakes?** \ No newline at end of file +## Some interesting properties + +The mean of the exponential distribution can be obtained integrating by parts as + +$ +E[X] = \int_o^{\infty}{x e^{-\lambda x}dx} = [-xe^{-\lambda x}]_0^{\infty} + \int_0^{\infty}{e^{-\lambda x}dx}= 1/ \lambda +$ + +As previously mentioned, the Exponential distribution models the waiting time between events (e.g.: floods or earthquakes) of a Poisson process. For the Poisson process, $\lambda$ is the rate at which the events occur. Therefore, $1/\lambda$ is the average time between those events, since it is the expectation of the associated Exponential distribution. In reliability analysis, it can be called *mean life time* or *time to failure*. + +The variance is given as + +$ +Var[X] = E[X^2]-(E[X])^2 = 1/\lambda^2 +$ + +It should be noted that the coefficient of variation $CV = \sqrt{Var[X]}/E[X] = 1$. + +If you remember the Poisson process is characterized by independent events. Thus, the time of the next occurrence modelled by the Exponential distribution is independent of the present and past occurrences. This is called the *memoryless property of the exponential distribution*. We can show it calculating the probability of the waiting time until the next ocurrence $X$ being higher than the sum of the two previous ones $X>x_1+x_2$, given that it is already bigger than one of them $X>x_1$ (conditional probability) as follows + +$ +P[X>x_1+x_2|X>x_1]=\frac{P[X>x_1+x_2]}{P[X>x_1]}=\frac{e^{\lambda (x_1+x_2)}}{e^{\lambda x_1}}=e^{\lambda x_2}=P[X>x_2] +$ + +If the waiting time $X$ is exponential-distributed, you can see that the probability of the waiting time of one occurence ($x_2$) does not depend on the previous one ($x_1$). + +## Let's practice! + -- GitLab From a353d0845fc985ff718bc2e5b403e3d10078fe90 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 11:36:38 +0200 Subject: [PATCH 06/21] more exponential --- book/sandbox/continuous/Exponential.md | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/book/sandbox/continuous/Exponential.md b/book/sandbox/continuous/Exponential.md index 389aed04..8bfa5af8 100644 --- a/book/sandbox/continuous/Exponential.md +++ b/book/sandbox/continuous/Exponential.md @@ -65,7 +65,4 @@ $ P[X>x_1+x_2|X>x_1]=\frac{P[X>x_1+x_2]}{P[X>x_1]}=\frac{e^{\lambda (x_1+x_2)}}{e^{\lambda x_1}}=e^{\lambda x_2}=P[X>x_2] $ -If the waiting time $X$ is exponential-distributed, you can see that the probability of the waiting time of one occurence ($x_2$) does not depend on the previous one ($x_1$). - -## Let's practice! - +If the waiting time $X$ is exponential-distributed, you can see that the probability of the waiting time of one occurence ($x_2$) does not depend on the previous one ($x_1$). \ No newline at end of file -- GitLab From d89984f3bc08577d7762bc50412bfb448661531e Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 12:25:36 +0200 Subject: [PATCH 07/21] first exponential example --- book/_toc.yml | 1 + book/sandbox/continuous/Exponential_quiz.md | 38 +++++++++++++++++++++ 2 files changed, 39 insertions(+) create mode 100644 book/sandbox/continuous/Exponential_quiz.md diff --git a/book/_toc.yml b/book/_toc.yml index 9dab1a0d..8a103fbe 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -25,6 +25,7 @@ parts: - file: sandbox/continuous/other_distr.md - file: sandbox/continuous/Uniform.md - file: sandbox/continuous/Exponential.md + - file: sandbox/continuous/Exponential_quiz.md - file: sandbox/continuous/Parameterization.md - file: sandbox/continuous/fitting.md - file: sandbox/continuous/GOF diff --git a/book/sandbox/continuous/Exponential_quiz.md b/book/sandbox/continuous/Exponential_quiz.md new file mode 100644 index 00000000..83cb719b --- /dev/null +++ b/book/sandbox/continuous/Exponential_quiz.md @@ -0,0 +1,38 @@ + +# Exponential distribution: let's practice + +An engineer is the manager of the construction of a bridge across a river and is concerned about large floods (floods with discharges higher than $50 m^3/s$) in the river, since they may damage the construction works. Higher floods than that occur in average every 10 years. + +Assume that floods are independent and identically distributed. This means that they can be modelled using a Poisson process and, thus, the time between floods can be modelled using an Exponential distribution. + + +What is the rate ($\lambda$) of the Exponential distribution? + + +```{admonition} Answer +:class: tip, dropdown + +The rate ($\lambda$) represents the frequency which is the inverse of the average waiting time. Therefore: + +$$ + \lambda = 1/10 = 0.1 +$$ + +``` + + +The construction time is planned to be 2 years. Under the previous hypothesis, what is the probability of not observing a flood in those 2 years? + + +```{admonition} Answer +:class: tip, dropdown + +The probability of not observing a flood in two years can be computed as the probability of the waiting time between floods being higher than 2 years. Therefore: + +$$ + P[X>2] = 1 - F(x) = 1 - (1 - e^{-0.1 \cdot 2}) = e^{-0.1 \cdot 2} \approx 0.82 +$$ + +This means that there is a probability of observing a flood of 1 - 0.82 = 0.18, which is actually pretty high! Maybe the engineer should think about a tighter schedule or using some adaptive techniques so the construction works are not affected by events that low. + +``` \ No newline at end of file -- GitLab From ca52e2563e36e8e9afa3f071c0038f8e9e0d6380 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 13:37:14 +0200 Subject: [PATCH 08/21] start fitting --- book/_toc.yml | 1 + book/sandbox/continuous/fitting.md | 17 ++++------------- book/sandbox/continuous/lmoments.md | 4 ++++ 3 files changed, 9 insertions(+), 13 deletions(-) create mode 100644 book/sandbox/continuous/lmoments.md diff --git a/book/_toc.yml b/book/_toc.yml index 8a103fbe..79d01c9f 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -28,6 +28,7 @@ parts: - file: sandbox/continuous/Exponential_quiz.md - file: sandbox/continuous/Parameterization.md - file: sandbox/continuous/fitting.md + - file: sandbox/continuous/lmoments.md - file: sandbox/continuous/GOF - file: sandbox/SanPart/ObservationTheory/01_Introduction.md title: Sensing and Observation Theory diff --git a/book/sandbox/continuous/fitting.md b/book/sandbox/continuous/fitting.md index cc0aad0c..4c4a8129 100644 --- a/book/sandbox/continuous/fitting.md +++ b/book/sandbox/continuous/fitting.md @@ -1,16 +1,7 @@ -# Fitting a Distribtuion +# Fitting a Distribution -In the previous section, distr. +In the previous sections, we introduced parametric distributions as mathematical models for the empirical distribution functions of observations. Also, some of the most widely used parametric distributions were presented, such as Exponential or Normal distribution. Those parametric distributions had some parameters which required to be fitted to the observations. For instance, Exponential distribution has a rate parameter ($\lambda$). In this section, the most commonly used methods to fit the parametric distribution functions are presented. -## L-moments - -moments distr = moments data - -## MLE - -MLE video - -## Quizz/exercises - -practice \ No newline at end of file +- L-moments +- Maximum loglikelihood estimator (MLE) diff --git a/book/sandbox/continuous/lmoments.md b/book/sandbox/continuous/lmoments.md new file mode 100644 index 00000000..2a470e10 --- /dev/null +++ b/book/sandbox/continuous/lmoments.md @@ -0,0 +1,4 @@ + +# Fitting a Distribution: L-moments + +In the previous sections, -- GitLab From 00eb9e37683dc9a33a3bddff5b6e11f041f073d6 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 14:06:30 +0200 Subject: [PATCH 09/21] start gumbel and moments --- book/sandbox/continuous/Gumbel.md | 13 +++++++++++++ book/sandbox/continuous/fitting.md | 4 ++-- book/sandbox/continuous/moments.md | 6 ++++++ 3 files changed, 21 insertions(+), 2 deletions(-) create mode 100644 book/sandbox/continuous/Gumbel.md create mode 100644 book/sandbox/continuous/moments.md diff --git a/book/sandbox/continuous/Gumbel.md b/book/sandbox/continuous/Gumbel.md new file mode 100644 index 00000000..5f0cebee --- /dev/null +++ b/book/sandbox/continuous/Gumbel.md @@ -0,0 +1,13 @@ + +# Gumbel distribution + +Gumbel distribution is widely used to model maximum and minimum values of natural phenomena, such as wave storms or floods. You will learn more about this application in the Extreme Value Analysis section. The PDF of the Gumbel distribution is given by + +$ +f(x) = \frac{1}{\beta}e^{-\left( \frac{x-\mu}{\beta} + e^{-\left( \frac{x-\mu}{\beta} \right)}\right)} +$ + +where $\mu$ is the location parameter and $\beta>0$ is the scale parameter. + + + diff --git a/book/sandbox/continuous/fitting.md b/book/sandbox/continuous/fitting.md index 4c4a8129..a7bfc762 100644 --- a/book/sandbox/continuous/fitting.md +++ b/book/sandbox/continuous/fitting.md @@ -1,7 +1,7 @@ # Fitting a Distribution -In the previous sections, we introduced parametric distributions as mathematical models for the empirical distribution functions of observations. Also, some of the most widely used parametric distributions were presented, such as Exponential or Normal distribution. Those parametric distributions had some parameters which required to be fitted to the observations. For instance, Exponential distribution has a rate parameter ($\lambda$). In this section, the most commonly used methods to fit the parametric distribution functions are presented. +In the previous sections, we introduced parametric distributions as mathematical models for the empirical distribution functions of observations. Also, some of the most widely used parametric distributions were presented, such as Exponential or Normal distribution. Those parametric distributions had some parameters which required to be fitted to the observations. For instance, Exponential distribution has a rate parameter ($\lambda$). In this section, the most commonly used methods to fit the parametric distribution functions are presented. Note that it assumes that we know the parametric distribution we want to fit. For instance, fitting an Exponential distribution to a sample of observations. -- L-moments +- L-moments or Method of moments - Maximum loglikelihood estimator (MLE) diff --git a/book/sandbox/continuous/moments.md b/book/sandbox/continuous/moments.md new file mode 100644 index 00000000..7bbc78d7 --- /dev/null +++ b/book/sandbox/continuous/moments.md @@ -0,0 +1,6 @@ + +# Fitting a Distribution: Method of moments + +The method of moments consists of equating the moments in the observations to those of the distribution we want to fit. Actually, you have already done it in the exercises of the Exponential distribution! Let's see it with another example. + +aaa -- GitLab From efa90180a941a2ce02b65e0128287b253a100d26 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 14:57:58 +0200 Subject: [PATCH 10/21] finish gumbel and modify exponential --- book/sandbox/continuous/Exponential.md | 4 ++- book/sandbox/continuous/Gumbel.md | 44 +++++++++++++++++++++++++- 2 files changed, 46 insertions(+), 2 deletions(-) diff --git a/book/sandbox/continuous/Exponential.md b/book/sandbox/continuous/Exponential.md index 8bfa5af8..5c18a043 100644 --- a/book/sandbox/continuous/Exponential.md +++ b/book/sandbox/continuous/Exponential.md @@ -65,4 +65,6 @@ $ P[X>x_1+x_2|X>x_1]=\frac{P[X>x_1+x_2]}{P[X>x_1]}=\frac{e^{\lambda (x_1+x_2)}}{e^{\lambda x_1}}=e^{\lambda x_2}=P[X>x_2] $ -If the waiting time $X$ is exponential-distributed, you can see that the probability of the waiting time of one occurence ($x_2$) does not depend on the previous one ($x_1$). \ No newline at end of file +If the waiting time $X$ is exponential-distributed, you can see that the probability of the waiting time of one occurence ($x_2$) does not depend on the previous one ($x_1$). + +Finally, note that this distribution is not bell-shaped and, thus, not symmetric. It presents a positive tail and its skewness is 2. \ No newline at end of file diff --git a/book/sandbox/continuous/Gumbel.md b/book/sandbox/continuous/Gumbel.md index 5f0cebee..3fb9449e 100644 --- a/book/sandbox/continuous/Gumbel.md +++ b/book/sandbox/continuous/Gumbel.md @@ -7,7 +7,49 @@ $ f(x) = \frac{1}{\beta}e^{-\left( \frac{x-\mu}{\beta} + e^{-\left( \frac{x-\mu}{\beta} \right)}\right)} $ -where $\mu$ is the location parameter and $\beta>0$ is the scale parameter. +where $\mu$ is the location parameter and $\beta>0$ is the scale parameter. In the figure below the influence of these parameters is presented. +In the right pannel of the figure below, an example of the PDF of three Gumbel Exponential distributions is shown. The first two distributions (black and blue continuous lines) present the same $\beta=1$ but different $\mu_{black}=0$ and $\mu_{blue}=2$. This produces a shift towards the positive values of the distribution, being the mode $=\mu$. Note that the shape of the PDF is identical, so $\mu$ does not influence the dispersion of the distribution. This can also be seen in the left pannel, where random samples from the distribution are drawn. The samples coming from the distribution with $\mu_{blue}=2$ are higher than those coming from the one with $\mu_{black}=0$, while both present a similar dispersion. +Regarding the third distribution (red crosses in the left pannel and red dashed line in the right pannel), it presents the same $\mu=0$ than the first distribution (black dots in the left pannel and continuous black line in the right pannel) but different $\beta$: $\beta_{black}=1$ and $\beta_{red}=3$. In the right pannel, it can be observed that both distributions present the same mode in $x=0$. However, the dispersion of the distribution with $\beta_{red}=3$ is way higher. This effect can also be seen in the random samples in the left pannel. Random samples from $\beta_{black}=1$ range from -2 to 4 approximately, while random samples from $\beta_{red}=3$ range from -2 to 8 approximately, having one sample a value of almost 14. +```{figure} /sandbox/continuous/figures/gumbel.png + +--- + +--- +Gumbel distribution function: (left) random samples, and (right) PDF. +``` + +Integrating the PDF, the following expression of the CDF is derived + +$ +F(x) = e^{-e^{-\frac{x-\mu}{\beta}}} +$ + +which is displayed in the figure below. + +```{figure} /sandbox/continuous/figures/gumbel_cdf.png + +--- + +Gumbel distribution function: CDF. +``` + +The influence of the parameters can also be observed in the CDF. The continuous blue and black distributions present the same shape while the blue distribution is shifted towards positive values due to the higher value of $\mu$. Regarding the dashed red distribution, the slope of the CDF is way more gentle, showing a higher dispersion than the other two distributions due to the higher value of $\beta$. + +## Some properties + +The mean of the Gumbel distribution can be computed as + +$ +E[X]=\mu + \gamma \beta +$ + +where $\gamma \approx 0.577$ is the Euler-Mascheroni constant. The variance is given by + +$ +Var[X] = \frac{\pi^2}{6}\beta^2 +$ + +Finally, note that Gumbel distribution is not symmetric and presents positive skewness. This is, it presents a tail towards positive values. 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Date: Tue, 1 Aug 2023 14:59:53 +0200 Subject: [PATCH 12/21] add Gumbel to TOC --- book/_toc.yml | 1 + 1 file changed, 1 insertion(+) diff --git a/book/_toc.yml b/book/_toc.yml index 79d01c9f..12e42d25 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -26,6 +26,7 @@ parts: - file: sandbox/continuous/Uniform.md - file: sandbox/continuous/Exponential.md - file: sandbox/continuous/Exponential_quiz.md + - file: sandbox/continuous/Gumbel.md - file: sandbox/continuous/Parameterization.md - file: sandbox/continuous/fitting.md - file: sandbox/continuous/lmoments.md -- GitLab From c463cc8922531465ce58aa2b8a9da30607b9b5fa Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 15:47:27 +0200 Subject: [PATCH 13/21] Gumbel exercise --- book/sandbox/continuous/Gumbel_quiz.md | 53 ++++++++++++++++++ .../figures/gumbel_concentration.png | Bin 0 -> 48481 bytes 2 files changed, 53 insertions(+) create mode 100644 book/sandbox/continuous/Gumbel_quiz.md create mode 100644 book/sandbox/continuous/figures/gumbel_concentration.png diff --git a/book/sandbox/continuous/Gumbel_quiz.md b/book/sandbox/continuous/Gumbel_quiz.md new file mode 100644 index 00000000..c2d2044b --- /dev/null +++ b/book/sandbox/continuous/Gumbel_quiz.md @@ -0,0 +1,53 @@ + +# Gumbel distribution: let's practice + +Groundwater quality of shallow aquifer has been tracked for a year to determine if it is duable to use it or if it is getting polluted. To this end, the daily maximum concentration of chloride in miligrams per liter has been measured. These concentration has been described using a Gumbel distribution with parameters $\mu=60$ and $\beta=10$. + +Chlorides between 1mg/L an 100mg/L are normal in freshwater. Use the Gumbel distribution to compute the probability of being in that interval. + +```{admonition} Answer +:class: tip, dropdown + +The probability of being below 100mg/L is: + +$$ + P[X \leq 100] = F(100) = e^{-e^{-\frac{100-60}{10}}} \approx 0.98 +$$ + +The same way, the probability of being below 1mg/L is: + +$$ + P[X > 1] = F(1) = e^{-e^{-\frac{1-60}{10}}} \approx 0 +$$ + +Thus, the probability of being in the interval (0,100) is + +$$ + P[1 \leq X \leq 100] = 0.98 - 0 = 0.98 +$$ + +The figure below illustrates the area to integrate in the PDF (blue area in the left pannel) and the probabilities computed in the CDF (right pannel). + +```{figure} /sandbox/continuous/figures/gumbel_concentration.png + +--- + +``` + + +And what is the probability of exceeding the upper limit of 100mg/L? + +```{admonition} Answer +:class: tip, dropdown + +Making use of the probability axioms: + +The probability of being below 100mg/L is: + +$$ + P[X>100]=1-P[X \leq 100] = 1-F(100) = 1- 0.98 = 0.02 +$$ + +This probability is very small, so the engineers can conclude that the water in the aquifer seems to present the characteristics of fresh water regarding the concentration of Chlorides. + +``` diff --git a/book/sandbox/continuous/figures/gumbel_concentration.png b/book/sandbox/continuous/figures/gumbel_concentration.png new file mode 100644 index 0000000000000000000000000000000000000000..0468aef701c9da3e65f2ebd1a4a715489398538c GIT binary patch literal 48481 zcmbTe30RKp+BSTbAyJxCq=7~nRcO$F=PnUNDiO^J(M<DTh(aZ4qQMYppb{$0Drpju zN~1=a2bza}Kl40mz3=<2_xrd1Z`*8ZJ#=69b)DC79>;#{`+l7M2etMvuH;%tp->q2 zYEX43l%=T@3eB$-%kei7AHT5Te-xb544e+zS~@wKI-H~&Fm<v!YwL9OwAqFWCmkG5 z+uF!VZj+P|-*C#w$<9$xO6uIdenHaK!AeSWJD(aZLT{&W)R97AG9`a#o~p*5rciP| z?4|C~b&2Y4c6QM{IsI;A=;i)5cXn=IzN-<O=)+R-c-_*k%O!6ZY<}~)<gH}#_3Oz` ze>@-LzcDYq=}=f|_8Ku7T8%@Bw5&VhG=50X{I#;oLMC6eouT;3t)0SF<#Sn*k|80E zADYHk14Xo3CKQCjE4p@5|Nbf76B)321OMMYI=t+u|N58sqXDa#PhG!$UEb*j^PxkB 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z{Waj--k{SSL94@oGk?HR8hB%T{!06czTV!=QEQI@7x@`feR)yT2dcwX7=T+Q+Q0#2 zpA&~#xk0O3K-0YIffx7gd<&e;-U%Gt1I|Us0T*R#sQ+KL26&-6@P=D^U^DRH%a@(N z3}qyn3>*Rj7I<R7qh%uf0zqv%P}-XaUYB^H1=vJ?3|v94`AO3}bzy);iT7t<gFUY5 zr7CEg54hIgUtr#G;3BpIz{4UFGEde=fgDw|p-jn0eoqT<6($7u%!+x%0ONFcf!5eV f0qS7*U;C~}qVdZ@O@LdH7#KWV{an^LB{Ts5{mN%6 literal 0 HcmV?d00001 -- GitLab From 0fa11aa0cb4af3c68fdf54746df1d7e1f480e1e4 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Tue, 1 Aug 2023 15:57:53 +0200 Subject: [PATCH 14/21] update toc --- book/_toc.yml | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/book/_toc.yml b/book/_toc.yml index 12e42d25..96e46242 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -27,9 +27,10 @@ parts: - file: sandbox/continuous/Exponential.md - file: sandbox/continuous/Exponential_quiz.md - file: sandbox/continuous/Gumbel.md + - file: sandbox/continuous/Gumbel_quiz.md - file: sandbox/continuous/Parameterization.md - file: sandbox/continuous/fitting.md - - file: sandbox/continuous/lmoments.md + - file: sandbox/continuous/moments.md - file: sandbox/continuous/GOF - file: sandbox/SanPart/ObservationTheory/01_Introduction.md title: Sensing and Observation Theory -- GitLab From 7873c8bbe0a521ae2b0d4e92493b2fc91c1eae4d Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Thu, 3 Aug 2023 11:06:27 +0200 Subject: [PATCH 15/21] properties uniform --- book/sandbox/continuous/Uniform.md | 18 +++++++++++++++++- 1 file changed, 17 insertions(+), 1 deletion(-) diff --git a/book/sandbox/continuous/Uniform.md b/book/sandbox/continuous/Uniform.md index 6300b6bc..f2458b78 100644 --- a/book/sandbox/continuous/Uniform.md +++ b/book/sandbox/continuous/Uniform.md @@ -34,4 +34,20 @@ $ F(x) = 0 \hspace{1cm} for \ x>b $ -If we make $a=0$ and $b=1$, we obtain the standard or unity uniform distribution, which is used to generate random values from other distribution functions for simulation purposes. \ No newline at end of file +If we make $a=0$ and $b=1$, we obtain the standard or unity uniform distribution, which is used to generate random values from other distribution functions for simulation purposes. + +## Some properties + +The mean of the uniform distribution can be computed based on its simple geometry as + +$ +E[X]=\frac{1}{2}(a+b) +$ + +The variance is given by + +$ +Var[X] = \frac{1}{12}(b-a)^2 +$ + +Finally, note that uniform distribution is symmetric and presents 0 skewness. This is, it does not present any tail. \ No newline at end of file -- GitLab From d8fb459f39fc52cefb05bb1466ba4021318ddf61 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Thu, 3 Aug 2023 11:06:53 +0200 Subject: [PATCH 16/21] fitting moments --- book/sandbox/continuous/moments.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/book/sandbox/continuous/moments.md b/book/sandbox/continuous/moments.md index 7bbc78d7..8be25162 100644 --- a/book/sandbox/continuous/moments.md +++ b/book/sandbox/continuous/moments.md @@ -1,6 +1,5 @@ # Fitting a Distribution: Method of moments -The method of moments consists of equating the moments in the observations to those of the distribution we want to fit. Actually, you have already done it in the exercises of the Exponential distribution! Let's see it with another example. +The method of moments consists of equating the moments in the observations to those of the distribution we want to fit. This is, there is a relationship between the moments and the parameters of the distribution. Then, if we equate the moments of the distribution to the moments of the observations, we can obtain the values of the parameters of the distribution. Actually, you have already done it in the exercises of the Exponential distribution! Let's see it with another example. -aaa -- GitLab From 2e1c57e6ccf042ef73ddc8ac3f834e1a7e9184b5 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Thu, 3 Aug 2023 11:23:50 +0200 Subject: [PATCH 17/21] typo Gumbel quiz --- book/sandbox/continuous/Gumbel_quiz.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/book/sandbox/continuous/Gumbel_quiz.md b/book/sandbox/continuous/Gumbel_quiz.md index c2d2044b..14740f26 100644 --- a/book/sandbox/continuous/Gumbel_quiz.md +++ b/book/sandbox/continuous/Gumbel_quiz.md @@ -1,7 +1,7 @@ # Gumbel distribution: let's practice -Groundwater quality of shallow aquifer has been tracked for a year to determine if it is duable to use it or if it is getting polluted. To this end, the daily maximum concentration of chloride in miligrams per liter has been measured. These concentration has been described using a Gumbel distribution with parameters $\mu=60$ and $\beta=10$. +Groundwater quality of a shallow aquifer has been tracked for a year to determine if it is duable to use it or if it is getting polluted. To this end, the daily maximum concentration of chloride in miligrams per liter has been measured. These concentration has been described using a Gumbel distribution with parameters $\mu=60$ and $\beta=10$. Chlorides between 1mg/L an 100mg/L are normal in freshwater. Use the Gumbel distribution to compute the probability of being in that interval. -- GitLab From bc0177dbbd706d15de4a2c8f5a98bf83fcc8e29f Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Thu, 3 Aug 2023 11:58:58 +0200 Subject: [PATCH 18/21] fitting with moments --- book/sandbox/continuous/moments.md | 75 +++++++++++++++++++++++++++++- 1 file changed, 74 insertions(+), 1 deletion(-) diff --git a/book/sandbox/continuous/moments.md b/book/sandbox/continuous/moments.md index 8be25162..6eb42a67 100644 --- a/book/sandbox/continuous/moments.md +++ b/book/sandbox/continuous/moments.md @@ -1,5 +1,78 @@ # Fitting a Distribution: Method of moments -The method of moments consists of equating the moments in the observations to those of the distribution we want to fit. This is, there is a relationship between the moments and the parameters of the distribution. Then, if we equate the moments of the distribution to the moments of the observations, we can obtain the values of the parameters of the distribution. Actually, you have already done it in the exercises of the Exponential distribution! Let's see it with another example. +The method of moments consists of equating the moments[^moment] in the observations to those of the distribution we want to fit. This is, there is a relationship between the moments and the parameters of the distribution. Then, if we equate the moments of the distribution to the moments of the observations, we can obtain the values of the parameters of the distribution. Actually, you have already done it in the exercises of the Exponential distribution! +**Let's see it with an example.** + +An engineer is studying the intensity of earthquakes in Rome (Italy). To this end, the engineer is using *Catalogo dei terremoti italiani dall'anno 1000 al 1980* (the Catalog of Italian earthquakes from year 1000 to 1980) edited by D. Postpischl in 1985. This catalog reports the intensity of earthquakes in terms of the Mercalli-Canconi-Sieber (MCS) index. Due to the uncertainties associated to this natural phenomenom, the engineer considers them as a random one and wants to fit a Gumbel distribution to the observations found in the catalog using the method of moments. The data found in the catalog is shown in the table below [^ref]. + + +```{list-table} MSC intensity of the recorded earthquakes in the city of Rome. +:header-rows: 1 +:name: earthquakes + +* - MSC intensity + - 2 + - 3 + - 4 + - 5 + - 6 + - 7 +* - Number + - 113 + - 132 + - 56 + - 22 + - 4 + - 2 +``` + +Remember that the CDF of the Gumbel distribution is given by + +$ +F(x) = e^{-e^{-\frac{x-\mu}{\beta}}} +$ + +Therefore, the value of $\mu$ and $\beta$ needs to be determined based on the observations to fit the distribution. + +The first thing the engineer needs to do is to calculate the moments of the observations in the Table. The mean ($\overline{X}$) and variance ($\sigma^2$) are calculated as + +$ +\overline{X} = \frac{2 \cdot 113 + 3 \cdot 132 + 4 \cdot 56 + 5 \cdot 22 + 6 \cdot 4 + 7 \cdot 2}{113+132+56+22+4+2} \approx 3.02 +$ + +$ +\sigma^2 = \frac{\sum{fx^2}}{\sum{f}}-\left( \frac{\sum{fx}}{\sum{f}}\right)^2 \approx 0.99 +$ + +where $x$ is the value of the intensity and $f$ is the frequency of the value $x$. + +Based on the properties of the Gumbel distribution, we know + +$ +E[X]=\mu + \gamma \beta +$ + +$ +Var[X] = \frac{\pi^2}{6}\beta^2 +$ + +where $\gamma \approx 0.577$ is the Euler-Mascheroni constant. + +Therefore, we can equate the expectation and variance of the distribution ($E[X]$ and $Var[X]$) to the calculated moments and obtain the value of the parameters as + +$ +3.02 = \mu + 0.577 \beta +$ + +$ +0.99 = \frac{\pi^2}{6}\beta^2 +$ + +Thus, $\mu \approx 2.57$ and $\beta \approx 0.77$. + + + +[^moment]: We denote as moments in statistics to the quantitative properties to characterize a distribution. The four commonly used moments are the mean, the variance, the skewness and the kurtosis. +[^ref]: Data extracted from Kottegoda and Rosso (2008). \ No newline at end of file -- GitLab From 92550ed34049050019f11fdef00de87df27728c2 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Thu, 3 Aug 2023 12:01:59 +0200 Subject: [PATCH 19/21] fitting with moments --- book/sandbox/continuous/moments.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/book/sandbox/continuous/moments.md b/book/sandbox/continuous/moments.md index 6eb42a67..b845559c 100644 --- a/book/sandbox/continuous/moments.md +++ b/book/sandbox/continuous/moments.md @@ -3,7 +3,7 @@ The method of moments consists of equating the moments[^moment] in the observations to those of the distribution we want to fit. This is, there is a relationship between the moments and the parameters of the distribution. Then, if we equate the moments of the distribution to the moments of the observations, we can obtain the values of the parameters of the distribution. Actually, you have already done it in the exercises of the Exponential distribution! -**Let's see it with an example.** +## Let's see it with an example An engineer is studying the intensity of earthquakes in Rome (Italy). To this end, the engineer is using *Catalogo dei terremoti italiani dall'anno 1000 al 1980* (the Catalog of Italian earthquakes from year 1000 to 1980) edited by D. Postpischl in 1985. This catalog reports the intensity of earthquakes in terms of the Mercalli-Canconi-Sieber (MCS) index. Due to the uncertainties associated to this natural phenomenom, the engineer considers them as a random one and wants to fit a Gumbel distribution to the observations found in the catalog using the method of moments. The data found in the catalog is shown in the table below [^ref]. -- GitLab From fede94dc38732e506154687bddc5ad6983abc694 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Thu, 3 Aug 2023 12:31:38 +0200 Subject: [PATCH 20/21] exercise moments --- book/sandbox/continuous/moments_quiz.md | 70 +++++++++++++++++++++++++ 1 file changed, 70 insertions(+) create mode 100644 book/sandbox/continuous/moments_quiz.md diff --git a/book/sandbox/continuous/moments_quiz.md b/book/sandbox/continuous/moments_quiz.md new file mode 100644 index 00000000..4b9027fd --- /dev/null +++ b/book/sandbox/continuous/moments_quiz.md @@ -0,0 +1,70 @@ + +# Method of moments: let's practice + +As part of the quality control of the construction of a building, lab tests are performed to determine the compressive strengths of concrete. The following values in $N/mm^2$ are obtained: 60.5, 59.8, 53.4, 56.9 and 61.9. + +The engineer responsible for quality assumes that the compressive strength follows a uniform distribution, whose CDF is given by + +$ +F(x) = 0 \hspace{1cm} for \ x<a +$ + +$ +F(x) = \frac{x-a}{b-a} \hspace{1cm} for \ a\leq x \leq b +$ + +$ +F(x) = 0 \hspace{1cm} for \ x>b +$ + +Determine the value of $a$ and $b$ based on the observations using the method of moments. + +```{admonition} Answer +:class: tip, dropdown + +The mean and variance of the observations can be calculates as: + +$$ + \overline{X} = 58.5 +$$ + + +$$ + \sigma^2 = 11.46 +$$ + +We know that the expectation and variance of the uniform distribution is given by + +$$ +E[X]=\frac{1}{2}(a+b) +$$ + +$$ +Var[X] = \frac{1}{12}(b-a)^2 +$$ + +Thus, equating the mean and variance in the observations to those of the distribution, we obtain the following system + +$$ +58.5=\frac{1}{2}(a+b) +$$ + +$$ +11.46 = \frac{1}{12}(b-a)^2 +$$ + +Solving the system, $a \approx 52.64$, and $b \approx 64.36$. Note that $a \approx 64.36$, and $b \approx 52.64$ is also a solution to the system, so we choose $a<b$ based on our definition of the uniform distribution. + +``` + +Now that the engineer knows the uniform distribution which characterizes the compressive strength of the concrete samples, he needs to compute the probability of being above $55N/mm^2$, since it is the required resistance for the construction. + +```{admonition} Answer +:class: tip, dropdown + +$$ +P[X>55N/mm^2] = 1 - F(55) = 1 - \frac{55-52.64}{64.36-52.64} \approx 0.80 +$$ + +Thus, there is a probability of 0.80 to fulfill the required resistence and a probabiltiy of 0.20 of not doing it. He will have to contact the contractor! +``` \ No newline at end of file -- GitLab From 84d00558b156d893fd97299edead3647e847e394 Mon Sep 17 00:00:00 2001 From: pmaresnasarre <p.maresnasarre@tudelft.nl> Date: Thu, 3 Aug 2023 12:33:41 +0200 Subject: [PATCH 21/21] add exercise moments to toc --- book/_toc.yml | 1 + book/sandbox/continuous/Uniform.md | 2 +- 2 files changed, 2 insertions(+), 1 deletion(-) diff --git a/book/_toc.yml b/book/_toc.yml index 96e46242..7a5a24e8 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -31,6 +31,7 @@ parts: - file: sandbox/continuous/Parameterization.md - file: sandbox/continuous/fitting.md - file: sandbox/continuous/moments.md + - file: sandbox/continuous/moments_quiz.md - file: sandbox/continuous/GOF - file: sandbox/SanPart/ObservationTheory/01_Introduction.md title: Sensing and Observation Theory diff --git a/book/sandbox/continuous/Uniform.md b/book/sandbox/continuous/Uniform.md index f2458b78..5ae0b8ba 100644 --- a/book/sandbox/continuous/Uniform.md +++ b/book/sandbox/continuous/Uniform.md @@ -50,4 +50,4 @@ $ Var[X] = \frac{1}{12}(b-a)^2 $ -Finally, note that uniform distribution is symmetric and presents 0 skewness. This is, it does not present any tail. \ No newline at end of file +Finally, note that uniform distribution is symmetric and presents 0 skewness. Thus, the median and the mean are identical. This is, it does not present any tail. \ No newline at end of file -- GitLab