discrete phase-type distribution

Imagine you have a bag of colored marbles. Each marble represents something that could happen, like a car breaking down or winning a prize. Now let's say you know how many marbles there are for each possibility, and how likely it is that you'll draw that marble out of the bag. That's kind of like having a discrete phase-type distribution.

Basically, a discrete phase-type distribution is a way of describing something that can happen in a certain number of steps or phases, where each step has a known probability of happening. It's like having a set of instructions for how something can happen, and knowing how likely each step is to occur.

For example, let's say you want to model how long it takes for a computer program to run. You know that there are a certain number of different steps that the program goes through, and each step takes a certain amount of time. With a discrete phase-type distribution, you can estimate how long the program will take by combining the probabilities of each step happening in the right order.

So, in summary, a discrete phase-type distribution is like having a bag of marbles or a set of instructions for how something happens, where each step has a known probability of happening. It's a way of organizing and predicting how things will play out, step by step.