Simulating a divide area random variable Suppose that we have a random variable with pdf like
$f = \left\{
\begin{array}{ c l }
x+1,    x\in [-1,0] \\
-x+1,   x\in (0,1]
\end{array}
\right.$ and we have to simulate it.
It's pdf is 
$F(x) = \left\{
\begin{array}{ c l }
x^2/2+x+0.5,    x\in [-1,0] \\
-x^2/2+x+0.5,   x\in (0,1]
\end{array}
\right.$
two methods to simulate this can be found here http://web.ics.purdue.edu/~hwan/IE680/Lectures/Chap08Slides.pdf page 12.
My question is if we can simulate problems like that by fist simulate a random U(0,1) and 
if it is smaller than the probability  $P(x\in [-1,0])=1/2$ simulate from the first part with inverse transformation $\sqrt{2U}-1$ but for a $U\sim Unif(0,1/2=P(x\in [-1,0]))$ 
and if it is bigger  simulate  with inverse transformation $1-\sqrt{(2(1-U))}$ using a $U\sim Unif(1/2,1)$. 
When I make the histogram it seems to working, but is it correct?
[code]rand=function(){
if(runif(1,0,1)<=1/2){x=sqrt(2*runif(1,0,1/2))-1}else{
x=1-sqrt(2-2*runif(1,1/2,1))
}
return(x)
}


pri=NULL


for(i in 1:10000)
{
pri=c(pri,rand())
}


hist(pri,prob=T)[/code]

 A: Yes, your code generates the correct results, but as you can see from the calculations below, it's hard to understand.  I suggest sticking with the linked lecture notes and use runif(1,0,1), which represents $\mathrm{Unif}(0,1)$.
The composition algorithm in the linked notes:

  
*
  
*Generate two independent $U_1, U_2 \sim \mathrm{Unif}(0,1)$
  
*
*
  
*If $U_1 < 1/2$,  return $X = \sqrt{U_2}-1$
  
*Otherwise, return $X = 1 - \sqrt{1-U_2}$
  
  

The composition algorithm in the question body contains a rescaled version of $U_2$.

  
*
  
*Generate $U_1 \sim \mathrm{Unif}(0,1)$
  
*
*
  
*$U_1 < 1/2$,  return $X = \sqrt{2U}-1$ for $U\sim \mathrm{Unif}(0,1/2)$
  In fact, $2U \stackrel{(d)}{=} U_2$.
  
*Otherwise, return $X = 1 - \sqrt{2-2U}$ for $U\sim \mathrm{Unif}(1/2,1)$
  In fact, $2U \sim \mathrm{Unif}(1,2)$, so $2-2U \sim \mathrm{Unif}(0,1)$ and $2-2U \stackrel{(d)}{=} 1-U_2$.
  
  


These figures illustrates the how the code works.

If we increase the sample size from 10k to 50k, the histogram looks more similar to the triangular distribution on $[-1,1]$.

We increasing the class number from 20 to 50 in order to take a closer look of the data.


My code
rand=function(){
if(runif(1,0,1)<=1/2){x=sqrt(2*runif(1,0,1/2))-1}else{
x=1-sqrt(2-2*runif(1,1/2,1))
}
return(x)
}

pri=NULL
for(i in 1:50000)
{
pri=c(pri,rand())
}

hist(pri,breaks=50,prob=T)

