Update toy document

module2/exo1/toy_document_orgmode_R_en.org

-#+TITLE:  Your title
-#+AUTHOR: Your name
-#+DATE:   Today's date
+#+TITLE: On the computation of pi
 #+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"/>
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+* Asking the maths library
 
-* Some explanations
+My computer tells me that $\pi$ is approximately
 
-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/.
+#+BEGIN_SRC R :results output :exports both
+pi
+#+END_SRC
 
-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.
+#+RESULTS:
+: [1] 3.141593
+
+* Buffon's needle
 
-Like we showed in the video, R code is included as follows (and is
-exxecuted by typing ~C-c C-c~):
+Applying the method of Buffon's needle, we get the *approximation*
 
-#+begin_src R :results output :exports both
-print("Hello world!")
-#+end_src
+#+BEGIN_SRC R :results output :exports both
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
+#+END_SRC
 
 #+RESULTS:
-: [1] "Hello world!"
+: [1] 3.14327
+
+* Using a surface fraction argument
 
-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~.
+A method that is easier to understand and does not make use of the sin
+function is based on the fact that if $X \sim U(0,1)$ and $Y \sim
+U(0,1)$, then $P[X^2 + Y^2 \le 1] = \pi/4$ (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 :session *R* :exports both
-summary(cars)
-#+end_src
+
+#+BEGIN_SRC R :session *R* :results output graphics :file "./pi.png" :exports both :width 600 :height 400
+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:
-:      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)
-#+end_src
+[[file:./pi.png]]
+
+It is then straightforward to obtain a (not really good) approximation
+to $\pi$ by counting how many times, on average, $X^2 + Y^2$ is smaller
+than 1:
+
+#+BEGIN_SRC R :session *R* :results output :exports both
+4*mean(df$Accept)
+#+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.
-
-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~.
-
-Now it's your turn! You can delete all this information and replace it
-by your computational document.
+: [1] 3.156