Generating Beta distributions with Uniform generators I can generate as many samples from one or more uniform distribution (0,1) as I wish. How can I use this to generate a beta distribution ?
 A: I will use [R] not so much as a practical answer (rbeta would do the trick), but as an attempt at thinking through the probability integral transform. I hope you are familiar with the code so you can follow, or replicate (if this answers your question).
The idea behind the Probability Integral Transform is that since a $cdf$ monotonically increases in value from $0$ to $1$, applying the $cdf$ function to random values form whichever distribution we may be interested in will on aggregate generate as many results say, between $0.1$ and $0.2$ as from $0.8$ to $0.9$. Now, this is exactly what a $pdf$ of a $U(0,1)$. It follows that if we start with values from a random uniform, $U \sim (0,1)$ instead, and we apply the inverse $cdf$ of the distribution we are aiming at, we'll end up with random values of that distribution.
Let's quickly show it with the queen of the distributions... The Normal $N(0,1)$. We generate $10,000$ random values, and plug them into the $erf$ function, plotting the results:
# Random variable from a normal distribution:
x <- rnorm(1e4)
par(mfrow=c(1,2))
hist(x, col='skyblue', main = "Random Normal")

# When transform by obtaining the cdf (x) will give us a Uniform:
y <- pnorm(x)
hist(y, col='skyblue', main = "CDF(X)")


In your case, we are aiming for $X \sim Beta(\alpha, \beta)$. So let's get started at the end and come up with $10,000$ random values from a $U(0,1)$. We also have to select values for the shape parameters of the $Beta$ distribution. We are not constrained there, so we can select for example, $\alpha=0.5$ and $\beta=0.5$. Now we are ready for the inverse, which is simply the qbeta function:
U <- runif(1e4)
alpha <- 0.5
beta <- 0.5
b_rand <- qbeta(U, alpha, beta)
hist(b_rand, col="skyblue", main = "Inverse U")

Compare this to the shape of the $Beta(\alpha,\beta)$ $pdf$:
x <- seq(0, 1, 0.001)
beta_pdf <- dbeta(x, alpha, beta)
plot(beta_pdf, type ="l", col='skyblue', lwd = 3,
     main = "Beta pdf")


A: You could use the inverse transform sampling method, which is useful to know about because it's a very general method that is not limited solely to the beta distribution. 
