Generate Beta distribution from Uniform random variables

Question

I need to generate random numbers from Beta distribution using random variables from Uniform distribution.
If I have two random variables $Y_1=U_1^{1/\alpha}$ and $Y_2=U_1^{1/\beta}$,
and If $Y_1+Y_2<= 1$ then $X=Y_1/(Y_1+Y_2)$ is from Beta distribution with parameters $\alpha\ and\ \beta$.

This is what I've done so far:
I've generated 10,000 random variables from a Uniform distribution.
I then used those rv to generate $Y_1 and\ Y_2 $ using $Y_1=U_1^{1/\alpha}$ and $Y_2=U_1^{1/\beta}$.
Then I generated $X$ using the formula $X=Y_1/(Y_1+Y_2)$.

Here is the R code I used:

library(dplyr)
# my plot
alpha <- 2
beta <- 4

set.seed(10)
u <- runif(10000,0,1)

y1 <- u^(1/alpha)
y2 <- u^(1/beta)


x <- data.frame(y1,y2 ) %>% 
  filter(y1+y2<=1) %>% # check that y1+y1 <=1
  mutate(x = y1/(y1+y2)) %>% # generate x random variable
  select(x) 

# plot x
hist(x$x)

# plot from beta distribution
hist(rbeta(10000,shape = 2,shape2 =4)) 

But when I plot this, the histogram seems to be going in the opposite direction to when I just generate rv from Beta distribution. Here is my plot:

And here is the plot if I generate rv from actual beta distribution:

Why does my curve look different to the one from Beta distribution?
This question seems very similar, but the solution uses qbeta() function. I think I need to use my $X=Y_1/(Y_1+Y_2)$

Answer 1

I just had the same problem with the distribution creation, thanks for the latest reply to the original post. Please find below a viable solution to create one RV in Python:

def beta(a,b):
    rv1 = np.random.rand()**(1/a)
    rv2 = np.random.rand()**(1/b)

    while (rv1+rv2) > 1:
        rv1 = np.random.rand()**(1/a)
        rv2 = np.random.rand()**(1/b)

    return = rv1 / (rv1+rv2)