Update toy_document_orgmode_R_en.org

module2/exo1/toy_document_orgmode_R_en.org

-#+TITLE:  Your title
-#+AUTHOR: Your name
-#+DATE:   Today's date
-#+LANGUAGE: en
-# #+PROPERTY: header-args :eval never-export
-
-#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/htmlize.css"/>
-#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/readtheorg.css"/>
-#+HTML_HEAD: <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
-#+HTML_HEAD: <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js"></script>
-#+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/lib/js/jquery.stickytableheaders.js"></script>
-#+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/readtheorg/js/readtheorg.js"></script>
-
-* Some explanations
-
-This is an org-mode document with code examples in R.  Once opened in
-Emacs, this document can easily be exported to HTML, PDF, and Office
-formats. For more information on org-mode, see
-https://orgmode.org/guide/.
-
-When you type the shortcut =C-c C-e h o=, this document will be
-exported as HTML. All the code in it will be re-executed, and the
-results will be retrieved and included into the exported document. If
-you do not want to re-execute all code each time, you can delete the #
-and the space before ~#+PROPERTY:~ in the header of this document.
-
-Like we showed in the video, R code is included as follows (and is
-exxecuted by typing ~C-c C-c~):
-
-#+begin_src R :results output :exports both
-print("Hello world!")
+# -*- coding: utf-8 -*-
+#+TITLE:       On the computation of pi
+#+AUTHOR:      Jessica Dagostini
+#+HTML_HEAD: <link rel="stylesheet" title="Standard" href="http://orgmode.org/worg/style/worg.css" type="text/css" />
+#+STARTUP: overview indent inlineimages logdrawer
+#+LANGUAGE:    en
+#+PROPERTY: header-args :eval never-export
+#+EXPORT_SELECT_TAGS: Blog
+#+OPTIONS:   H:2 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
+#+OPTIONS:   TeX:t LaTeX:nil skip:nil d:nil todo:t pri:nil tags:not-in-toc
+
+* Table of Contents
+ - [[Asking the maths library][1. Asking the maths library]]
+ - [[Buffon's needle][2. Buffon's needle]]
+ - [[Using a surface fraction argument][3. Using a surface frcation argument]]
+
+* Asking the maths library
+My computer tells me that \pi is /approximatively/
+#+begin_src R :results output :session *R* :exports both
+pi
 #+end_src
 
 #+RESULTS:
-: [1] "Hello world!"
-
-And now the same but in an R session. This is the most frequent
-situation, because R is really an interactive language. With a
-session, R's state, i.e. the values of all the variables, remains
-persistent from one code block to the next. The code is still executed
-using ~C-c C-c~.
+: [1] 3.141593
 
+* Buffon's needle
+Applying the method of [[https://en.wikipedia.org/wiki/Buffon's_needle_problem][Buffon's needle]], we get the *aprproximation*
 #+begin_src R :results output :session *R* :exports both
-summary(cars)
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
 #+end_src
 
 #+RESULTS:
-:      speed           dist       
-:  Min.   : 4.0   Min.   :  2.00  
-:  1st Qu.:12.0   1st Qu.: 26.00  
-:  Median :15.0   Median : 36.00  
-:  Mean   :15.4   Mean   : 42.98  
-:  3rd Qu.:19.0   3rd Qu.: 56.00  
-:  Max.   :25.0   Max.   :120.00
-
-Finally, an example for graphical output:
-#+begin_src R :results output graphics :file "./cars.png" :exports results :width 600 :height 400 :session *R* 
-plot(cars)
+: [1] 3.14327
+
+* Using a surface fraction argument
+A method that is easier to understand and does not make use of the sin
+function is based on the fact that if $X ∼ U(0,1)$ and $Y ~ U(0,1)$,
+then $P[X^2 + Y^2 \leq 1] = \pi/4$ (see [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" on
+Wikipedia]]). The following code uses this approach:
+#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R* 
+set.seed(42)
+N = 1000
+df = data.frame(X = runif(N), Y = runif(N))
+df$Accept = (df$X**2 + df$Y**2 <=1)
+library(ggplot2)
+ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()
 #+end_src
 
 #+RESULTS:
-[[file:./cars.png]]
-
-Note the parameter ~:exports results~, which indicates that the code
-will not appear in the exported document. We recommend that in the
-context of this MOOC, you always leave this parameter setting as
-~:exports both~, because we want your analyses to be perfectly
-transparent and reproducible.
-
-Watch out: the figure generated by the code block is /not/ stored in
-the org document. It's a plain file, here named ~cars.png~. You have
-to commit it explicitly if you want your analysis to be legible and
-understandable on GitLab.
+[[file:/tmp/babel-12624X4Q/figure12624Gry.png]]
 
-Finally, don't forget that we provide in the resource section of this
-MOOC a configuration with a few keyboard shortcuts that allow you to
-quickly create code blocks in R by typing ~<r~ or ~<R~ followed by
-~Tab~.
+It is then straightforward to obtain a (not really good) approximation
+to \pi by counting how many times, on average, $X² + Y²$ is smaller then
+1:
+#+begin_src R :results output :session *R* :exports both
+4*mean(df$Accept)
+#+end_src
 
-Now it's your turn! You can delete all this information and replace it
-by your computational document.
+#+RESULTS:
+: [1] 3.156