Commit c8d941c5 authored by Alexandre Jesus's avatar Alexandre Jesus

Minor fixes

parent daf6d9d6
...@@ -5,37 +5,36 @@ ...@@ -5,37 +5,36 @@
My computer tells me that $\pi$ is approximately My computer tells me that $\pi$ is approximately
#+BEGIN_SRC R :results output :exports both #+begin_src R :results output :exports both
pi pi
#+END_SRC #+end_src
#+RESULTS: #+RESULTS:
: [1] 3.141593 : [1] 3.141593
* Buffon's needle * Buffon's needle
Applying the method of Buffon's needle, we get the *approximation* Applying the method of [[https://en.wikipedia.org/wiki/Buffon%2527s_needle_problem][Buffon's needle]], we get the *approximation*
#+BEGIN_SRC R :results output :exports both #+begin_src R :results output :exports both
set.seed(42) set.seed(42)
N = 100000 N = 100000
x = runif(N) x = runif(N)
theta = pi/2*runif(N) theta = pi/2*runif(N)
2/(mean(x+sin(theta)>1)) 2/(mean(x+sin(theta)>1))
#+END_SRC #+end_src
#+RESULTS: #+RESULTS:
: [1] 3.14327 : [1] 3.14327
* Using a surface fraction argument * Using a surface fraction argument
A method that is easier to understand and does not make use of the sin A method that is easier to understand and does not make use of the
function is based on the fact that if $X \sim U(0,1)$ and $Y \sim $\sin$ function is based on the fact that if $X \sim U(0,1)$ and $Y
U(0,1)$, then $P[X^2 + Y^2 \le 1] = \pi/4$ (see [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" \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
on Wikipedia]]). The following code uses this approach method" on Wikipedia]]). The following code uses this approach
#+begin_src R :session *R* :results output graphics :file "./pi.png" :exports both :width 600 :height 400
#+BEGIN_SRC R :session *R* :results output graphics :file "./pi.png" :exports both :width 600 :height 400
set.seed(42) set.seed(42)
N = 1000 N = 1000
df = data.frame(X = runif(N), Y = runif(N)) df = data.frame(X = runif(N), Y = runif(N))
...@@ -45,7 +44,7 @@ ggplot(df, aes(x=X,y=Y,color=Accept)) + ...@@ -45,7 +44,7 @@ ggplot(df, aes(x=X,y=Y,color=Accept)) +
geom_point(alpha=.2) + geom_point(alpha=.2) +
coord_fixed() + coord_fixed() +
theme_bw() theme_bw()
#+END_SRC #+end_src
#+RESULTS: #+RESULTS:
[[file:./pi.png]] [[file:./pi.png]]
...@@ -54,9 +53,9 @@ It is then straightforward to obtain a (not really good) approximation ...@@ -54,9 +53,9 @@ 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 to $\pi$ by counting how many times, on average, $X^2 + Y^2$ is smaller
than 1: than 1:
#+BEGIN_SRC R :session *R* :results output :exports both #+begin_src R :session *R* :results output :exports both
4*mean(df$Accept) 4*mean(df$Accept)
#+END_SRC #+end_src
#+RESULTS: #+RESULTS:
: [1] 3.156 : [1] 3.156
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment