### INVERSE INTEGRAL TRANSFORM SAMPLING METHOD:

This is the answer to the original question posted in CV:

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?

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")

References:

Basic Inferential Calculations with only Summary Statistics