18-02. Stochastic simulations - How to simulate discrete random variables?

In this video, we show how to generate a discrete random variable (based on its probability mass function) from the uniform random variable on the unit interval, which is then illustrated in a computer program simulating the Bernoulli random variable. This is Section 17.1 of my Stochastic Modeling book.

[ojhurtado]

I do not understand. A uniform distribution is defined by two values, a and b, with a less than b. What does U{0, 1, ...2^n-1} mean then?
On the other hand the pdf(x)= 1/(b-a) always. And the cdf=(x-a)/(b-a) in the region.
So what does pk mean? X=k what values does it take?