Submit exercise 1

module2/exo1/toy_document_orgmode_R_en.org

-#+TITLE:  Your title
-#+AUTHOR: Your name
-#+DATE:   Today's date
+#+TITLE:  On the computation of pi
+#+AUTHOR: Jamal KHAN
+#+DATE: 2020-09-13
 #+LANGUAGE: en
 # #+PROPERTY: header-args :eval never-export
 
@@ -11,97 +11,51 @@
 #+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 <r tab (and is
-exxecuted by typing ~C-c C-c~):
-
-#+begin_src R :results output :exports both
-print("Hello world!")
+* Asking the maths library
+My computer tells me that \pi is approximattively
+#+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 **approximation**
 #+begin_src R :results output :session *R* :exports both
-summary(mtcars)
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
 #+end_src
 
 #+RESULTS:
-#+begin_example
-      mpg             cyl             disp             hp       
- Min.   :10.40   Min.   :4.000   Min.   : 71.1   Min.   : 52.0  
- 1st Qu.:15.43   1st Qu.:4.000   1st Qu.:120.8   1st Qu.: 96.5  
- Median :19.20   Median :6.000   Median :196.3   Median :123.0  
- Mean   :20.09   Mean   :6.188   Mean   :230.7   Mean   :146.7  
- 3rd Qu.:22.80   3rd Qu.:8.000   3rd Qu.:326.0   3rd Qu.:180.0  
- Max.   :33.90   Max.   :8.000   Max.   :472.0   Max.   :335.0  
-      drat             wt             qsec             vs        
- Min.   :2.760   Min.   :1.513   Min.   :14.50   Min.   :0.0000  
- 1st Qu.:3.080   1st Qu.:2.581   1st Qu.:16.89   1st Qu.:0.0000  
- Median :3.695   Median :3.325   Median :17.71   Median :0.0000  
- Mean   :3.597   Mean   :3.217   Mean   :17.85   Mean   :0.4375  
- 3rd Qu.:3.920   3rd Qu.:3.610   3rd Qu.:18.90   3rd Qu.:1.0000  
- Max.   :4.930   Max.   :5.424   Max.   :22.90   Max.   :1.0000  
-       am              gear            carb      
- Min.   :0.0000   Min.   :3.000   Min.   :1.000  
- 1st Qu.:0.0000   1st Qu.:3.000   1st Qu.:2.000  
- Median :0.0000   Median :4.000   Median :2.000  
- Mean   :0.4062   Mean   :3.688   Mean   :2.812  
- 3rd Qu.:1.0000   3rd Qu.:4.000   3rd Qu.:4.000  
- Max.   :1.0000   Max.   :5.000   Max.   :8.000
-#+end_example
-
-Finally, an example for graphical output:
-#+begin_src R :results output graphics :file "./cars.png" :exports results :width 600 :height 400 :session *R* 
+: 
+: [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 $X\simU(0,1)$ and $Y\simU(0,1)$, then
+$P[X^2+Y^2 \le1]=\pi/4$ (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() + 
-    geom_point(data=mtcars, mapping=aes(x=wt, y=mpg, color=factor(gear))) +
-    scale_x_continuous("Weight") +
-    scale_y_continuous("Milage") + 
-    scale_color_discrete("Gear") + 
-    theme_bw()
-
-ggsave('./cars.png', width=5, height=2.5, units="in", dpi=120)
-
+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-9oTMJE/figure0FnJcH.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, $X^2+Y^2$is smaller than 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