L03.9 Reliability
Independence is a very useful property. Whenever it is true, we can break up complicated situations into simpler ones. In particular, we can do separate calculations for each piece of a given model and then combine the results. We're going to look at an application of this idea into the analysis of reliability of a system that consists of independent units. So we have a system that consists of a number, let's say, n, of units. And each one of the units can be "up" or "down". And it's going to be "up" with a certain probability pi. Furthermore, we will assume that unit failures are independent. Intuitively, what we mean is that failure of some of the units does not the affect the probability that some of the other units will fail. If we want to be more formal, we might proceed as follows. We could define an event Ui to be the event that the ith unit is "up". And then make the assumption that the events U1, U2, and so on u...